The answer is almost certainly no.
The question linked by divibisan regards the Senate's powers as given under the constitution in Article One Section Three Clause Six (emphasis mine):
The Senate shall have the sole Power to try all Impeachments. When sitting for that Purpose, they shall be on Oath or Affirmation. When the President of the United States is tried, the Chief Justice shall preside: And no Person shall be convicted without the Concurrence of two thirds of the Members present.
The Supreme Court, in the opinion on Nixon v. United States, says:
As noted above, that sentence provides that "[t]he Senate shall have the sole Power to try all Impeachments." We think that the word "sole" is of considerable significance. Indeed, the word "sole" appears only one other time in the Constitution-with respect to the House of Representatives' "sole Power of Impeachment." Art. I, § 2, cl. 5 (emphasis added). The commonsense meaning of the word "sole" is that the Senate alone shall have authority to determine whether an individual should be acquitted or convicted. The dictionary definition bears this out. "Sole" is defined as "having no companion," "solitary," "being the only one," and "functioning ... independently and without assistance or interference." Webster's Third New International Dictionary 2168 (1971). If the courts may review the actions of the Senate in order to determine whether that body "tried" an impeached official, it is difficult to see how the Senate would be "functioning ... independently and without assistance or interference."
Judicial involvement in impeachment proceedings, even if only for purposes of judicial review, is counterintuitive because it would eviscerate the "important constitutional check" placed on the Judiciary by the Framers. See id., No. 81, at 545. Nixon's argument would place final reviewing authority with respect to impeachments in the hands of the same body that the impeachment process is meant to regulate.
In addition to the textual commitment argument, we are persuaded that the lack of finality and the difficulty of fashioning relief counsel against justiciability. See Baker v. Carr, 369 U. S., at 210. We agree with the Court of Appeals that opening the door of judicial review to the procedures used by the Senate in trying impeachments would "expose the political life of the country to months, or perhaps years, of chaos." 290 U. S. App. D. C., at 427, 938 F. 2d, at 246. This lack of finality would manifest itself most dramatically if the President were impeached. The legitimacy of any successor, and hence his effectiveness, would be impaired severely, not merely while the judicial process was running its course, but during any retrial that a differently constituted Senate might conduct if its first judgment of conviction were invalidated. Equally uncertain is the question of what relief a court may give other than simply setting aside the judgment of conviction. Could it order the reinstatement of a convicted federal judge, or order Congress to create an additional judgeship if the seat had been filled in the interim?
The Clause's first sentence must ... be read as a grant of authority to the Senate to determine whether an individual should be acquitted or convicted, and the commonsense and dictionary meanings of the word "sole" indicate that this authority is reposed in the Senate alone.
A review of the Constitutional Convention's history and the contemporary commentary supports a reading of the constitutional language as deliberately placing the impeachment power in the Legislature, with no judicial involvement, even for the limited purpose of judicial review.
In one of the concurring minority opinions, Justice White states:
In a truly balanced system, impeachments tried by the Senate would serve as a means of controlling the largely unaccountable Judiciary, even as judicial review would ensure that the Senate adhered to a minimal set of procedural standards in conducting impeachment trials.
In short, textual and historical evidence reveals that the Impeachment Trial Clause was not meant to bind the hands of the Senate beyond establishing a set of minimal procedures.
Which to me shows important limitations on judicial review.
And finally, Justice Souter in his concurring minority opinion states:
One can, nevertheless, envision different and unusual circumstances that might justify a more searching review of impeachment proceedings. If the Senate were to act in a manner seriously threatening the integrity of its results, convicting, say, upon a coin toss, or upon a summary determination that an officer of the United States was simply" 'a bad guy,'" ante, at 239 (WHITE, J., concurring in judgment), judicial interference might well be appropriate. In such circumstances, the Senate's action might be so far beyond the scope of its constitutional authority, and the consequent impact on the Republic so great, as to merit a judicial response despite the prudential concerns that would ordinarily counsel silence. "The political question doctrine, a tool for maintenance of governmental order, will not be so applied as to promote only disorder."
This, again, gives life to claims that the Senate failed to "try" the accused, but doesn't extend the review beyond those grounds.
Now, compare this with Article One Section Two Clause Five:
The House of Representatives shall choose their Speaker and other Officers; and shall have the sole Power of Impeachment.
Unlike the Senate, there are no Constitutional requirements of the House in exercising their power of impeachment.
tl;dr: The Constitution gives the House of Representatives the sole power of impeachment, full stop.