Trump has issued a directive see page 2 that has been used as a reason by some of his (present and former) officials to not testify in the impeachment hearings so far, including Bolton and Mulvaney. Does this Trump directive instruct his staff to not testify in a putative Senate trial either (assuming impeachment gets that far)?
According to the legal complaint (asking the Judicial branch to adjudicate the conflict), the directive in question was a communication sent to Kupperman (Bolton's aide) by the White House counsel.
On October 25, 2019, the White House Counsel transmitted a letter to counsel for Plaintiff, asserting the “constitutional immunity of current and former senior advisors to the President” and instructing Plaintiff not to appear and testify in response to the subpoena. A copy of the letter is attached hereto as Exhibit B. The White House Counsel stated that the Office of Legal Counsel of the Department of Justice had “advised [him] that [Plaintiff] is absolutely immune from compelled congressional testimony with respect to matters related to his service as a senior adviser to the President.” Exhibit B at 1.
So given the wording "immune from compelled congressional testimony", it seems that the directive claims a very broad testimonial immunity, not restricted to the House of Representatives initial proceedings, so it is probably including the Senate trial as well, although not explicitly.
On the other hand, the letter also did include a specific prohibition to not appear at a particular hearing.
The White House Counsel informed Plaintiff’s counsel that “in order to protect the prerogatives of the Office of the President today and in the future, and in response to your request, the President directs Mr. Kupperman not to appear at the Committee’s scheduled hearing on Monday, October 28, 2019.” Exhibit B at 2.
The basis of the White House counsel claims (as explained in the complaint later) is the 1971 Rehnquist memo. As a Lawfare article details the history of that
The foundation of the doctrine of immunity is a statement by then-Assistant Attorney General William Rehnquist: “The President and his immediate advisers—that is, those who customarily meet with the President on a regular or frequent basis—should be deemed absolutely immune from testimonial compulsion by a congressional committee. They not only may not be examined with respect to their official duties, but they may not even be compelled to appear before a congressional committee.” Initially, the doctrine was justified as a matter of comity. As Assistant Attorney General Theodore Olson explained in 1982, “The President is a separate branch of government. He may not compel congressmen to appear before him. As a matter of separation of powers, Congress may not compel him to appear before it. The President’s close advisors are an extension of the President.” The executive branch found support for that statement in the Supreme Court’s conclusion that the immunity provided to members of Congress by the Speech and Debate Clause of the Constitution also provides immunity to congressional aides because those aides are the “alter egos” of the members. Because Congress may not compel the president to provide testimony, in the executive branch’s view, the same analysis applies to compelled testimony of presidential advisers.
Over time, the executive branch has expanded that position and explained more fully the basis for its immunity position. The most extensive public explication of the doctrine is in the 2014 OLC opinion concluding that David Simas, a senior adviser to President Obama, was immune from compliance with the House oversight committee’s subpoena. Although some have questioned whether the Obama administration took this position, the OLC opinion and the letter from White House Counsel Neil Eggleston to the committee make it clear that the administration believed “Mr. Simas is immune from congressional compulsion to testify on matters relating to his officials duties” and, accordingly, would not appear as the subpoena required.
The fact that this position has been asserted by administrations of both parties does not, of course, make it valid. Indeed, the only court to have addressed a claim of presidential adviser immunity has resoundingly rejected both the claimed absolute immunity and a qualified immunity. But the 2014 OLC opinion makes it clear the executive branch does not accept the analysis in that nonprecedential decision.
Also of note (given the confusion in the other answer)
Importantly, if executive privilege does not apply in an impeachment proceeding, then neither do any of the prophylactic doctrines that have developed to protect it, including the immunity of presidential advisers such as Kupperman. Much has been written about the doctrine of testimonial immunity, and the executive branch does consider it a separate and distinct doctrine from executive privilege. It may be a separate doctrine, but even its “founder,” William Rehnquist, understood it to be a prophylactic doctrine arising out of and reliant on the doctrine of executive privilege. [...]
In so doing, the executive branch has created a new prophylactic executive privilege that rarely, if ever, requires the president to assert privilege yet still renders Congress virtually impotent in oversight disputes when the executive branch plays hardball.
Note that the term "prophylactic executive privilege" seems idiosyncratic to this Lawfare writer; others may not use it when referring to the [Rehnquist] doctrine of immunity from compelled congressional testimony. As that article aslo [re]counts, the Trump White House has only resorted to the classic assertion of executive privilege once (in relation to a census question), but has used the "prophylactic" one many times.
In a related 2019 case, the OLC asserts that
The President does not waive an adviser’s immunity from compelled congressional testimony by authorizing disclosure of any particular information. The disclosure’s impact on executive privilege does not ultimately bear on any underlying immunity from compelled testimony.
As WaPo notes in yet another Trump-related case, the list of people the White House considers covered by immunity from testimony to congress is pretty broad. Furthermore, the Trump White House also considered that the judicial branch has no power to adjudicate such matters:
The letter sent to the House Judiciary Committee by the White House on the subject of Corey Lewandowski’s testimony makes for startling reading.
It claims immunity from congressional subpoena for the following list of people and classes of information:
Aides to the President Trump, of course, such as Kellyanne Conway, who the White House claims is “absolutely immune from compelled congressional testimony.”
People who aren’t aides and have never been aides but have advised the president, such as Lewandowski.
People who may not have been aides or advised the president, but have provided information to him “in connection with the discharge of his duties.”
Communications, not just between the president and his advisers, but between his advisers and anyone else, “relating to information or advice that will inform the discharge of the President’s responsibilities.”
By this standard, Kim Kardashian and her husband Kanye West, who advised the president on the fate of rapper A$AP Rocky after his arrest in Sweden, among other matters, could be considered as immune as Conway.
No president likes headline-grabbing congressional oversight. They all consider it unfair and many resist, stall and occasionally obstruct. But none have asserted such sweeping exemption from oversight by a coequal branch of government as Trump.
Moreover, there’s nothing Congress can do about the White House’s resistance, Trump’s lawyers argued in its Sept. 6 motion to dismiss a suit by the Ways and Means Committee in the U.S. District Court for the District of Columbia, because the federal courts are not empowered to rule in such a dispute.
Congress may not “conscript the Judiciary on its side of a dispute with the Executive Branch over a congressional demand for information,” Trump’s lawyers stated.