If found some data in a Congressional report from 2001, but even that is pretty dated:
Cumulatively, in 1989 the United States was a party to 890 treaties
and 5,117 executive agreements.
There's also a long trend in the increase of executive agreements vs treaties:
Accompanying the increase in international agreements was the
increase of international agreements other than treaties, that is,
agreements not submitted to the Senate. As the preceding table
shows, in the first 50 years of U.S. history, twice as many treaties
were concluded as executive agreements. In the 50-year period
from 1839 to 1889 a few more executive agreements than treaties
were concluded. In the 50-year period from 1889 to 1939 almost
twice as many executive agreements as treaties were concluded. In
the period since 1939 executive agreements have comprised more
than 90 percent of the international agreements concluded.
The growth in executive agreements may be accounted for by a
number of factors. A primary factor is the sheer increase in volume
of the amount of business and contacts between the United
States and other countries. Many observers believe it would be impractical
to submit every international agreement the United
States enters to the Senate as a treaty.
An executive agreement is usually much simpler to conclude or amend than a treaty. The Senate,
with an already heavy workload, would quickly find itself overburdened
if all international agreements, no matter how minor in
importance, were submitted to it for advice and consent.
And they have this detailed table of treaties vs executive agreements:
Also, the source of authority for the latter is detailed:
Most executive agreements are concluded under the authority of
a statute or prior treaty. In a wide variety of laws Congress has
authorized the executive branch to conclude international agreements
in fields including foreign aid, agriculture, and mutual security.
Similarly, the Senate has approved numerous treaties that implicitly
or explicitly authorized further agreement among the parties.
As an example, the executive branch has concluded numerous
defense and base agreements on the basis of the North Atlantic
Treaty and other security treaties. One study found that 88.3 percent
of international agreements reached between 1946 and 1972
were based at least partly on statutory authority; 6.2 percent on
treaties, and 5.5 percent solely on executive authority.
So basically, the common case nowadays is that Congress "pre-approves" many such executive agreements (which are treaties in international law, even if not in US law--that's also said in the same source) simply by giving the Executive the power to enter into them in specific areas. This common case even has a name: congressional-executive agreements. It started
in 1790 Congress empowered the President to
pay off the Revolutionary War debt by borrowing money from foreign
countries ‘‘upon terms advantageous to the United States’’ and
to conclude ‘‘such other contracts respecting the said debt as shall
be found for the interest of the said States.’’ [...]
Over the years,
Congress has authorized or sanctioned additional agreements concerning
a wide variety of subjects including, inter alia, the protection
of intellectual property rights, acquisition of territory, national
participation in various international organizations, foreign
trade, foreign military assistance, foreign economic assistance, atomic energy cooperation, and international fishery rights.
The subject matter diversity of congressional-executive agreements
is matched by the varying means by which Congress has authorized
the conclusion of such agreements. Thus, Congress has enacted
statutes providing authority in advance for the President to
negotiate with other nations on a particular matter. This authority
may be explicit, or, in the case of agreements concluded in conformity
with a generally enunciated congressional policy, implied
from the terms of the enactment. Legislative authorization for
congressional-executive agreements may also be effected by passage
of a statute following the negotiation of a concluded agreement.
Again, congressional approval may be explicit, or, implied,
as in the case of legislation appropriating funds to carry out participation
by the United States in an international organization.
In regulating the use of congressional-executive agreements,
Congress has specified in advance the general terms of negotiation
and conditioned the effectiveness of particular agreements
alternatively upon the enactment of implementing legislation,
upon the legislative adoption of an approving concurrent resolution
within a specified time following transmittal of the agreement to
Congress, or upon the failure of Congress to adopt a disapproving
concurrent or joint resolution within designated time periods. Furthermore, congressional approval of some agreements has been
accompanied by conditions. The President is presently required by at least one statute to select Members of Congress from specified
committees to serve as accredited advisers to American delegations
attending international conferences, meeting, and negotiating
sessions relating to trade agreements. Other legislation has required the President to consult with specified committees before
entering into trade agreements.
[...]
The increasing use of international agreements other than treaties
challenged the Senate to oversee that the executive agreement
process was not used when agreements should properly be submitted
to the Senate as treaties. Similarly, the increasing rise of
agreements requiring approval by Congress, while assuring a congressional
role, challenged the Senate to distinguish which types of
agreements required submission to the Senate under the traditional
treaty procedure.
Also multilateral agreements are more likely (compared to bilateral agreements) to be treaties (in US law) rather than executive agreements:
Although multilateral executive agreements being concluded outnumber
multilateral treaties, multilateral agreements form a far
larger proportion of treaties than of executive agreements. Of 415
treaties that the United States concluded from 1980 through 1999,
155 (37 percent) were multilateral; of 6,381 executive agreements,
294 (4.6 percent) were multilateral.
Still, even in the case of multilateral agreements, the ratio of executive agreements to (US-law) treaties is about 2:1.
The Senate actually has a list of pending treaties. Currently there are 44 on that list, including very old ones like the Vienna Convention (signed in 1970). It's alas not entirely straightforward to compare that number with with those in previous paragraphs because of how old the data above is, but clearly it's a pretty small fraction of the 700-1,500 treaties (depending on the time frame) that the Senate has ratified. Furthermore, looking at that list there is a significant time-break around position #20, where the older treaties are from around year 2000 or before, while the entries in the 2nd half of the list are much newer, post-2008 or so. In my rough estimation, about half of the treaties listed there are fairly "intractable" for the Senate; the rest looks a bit more like a "regular" backlog of treaties that might still pass.
