Westminster’s Scottish Affairs Committee, made up of MPs from Scotland, said it would wait for the outcome of the Scottish Parliament vote before it made a decision on launching its own inquiry.
In a statement, the committee said it had agreed Holyrood was the “most appropriate place” for a probe.
If a Westminster inquiry was set up, both Swinney and Nicola Sturgeon could be asked to give evidence under oath.
On Tuesday, the first minister’s spokesman claimed that a probe would “destroy the credibility” of the Scottish Affairs Committee and its convener, Labour MP Patricia Gibson.
He would not be drawn on whether the first minister would co-operate with a Westminster investigation, replying that Swinney would “decide how to respond when we see what is agreed to”.
People can be summoned to appear in front of a UK Parliament committee, though they are unlikely to face any serious sanctions for refusing to do so.
If someone rejects a formal request, they can be found in contempt of parliament. Historically, those found guilty of contempt could be fined or imprisoned, though no-one has been locked up since 1880, or fined since 1666.
The Scottish Affairs Committee is made up of six Labour MPs, two Conservatives, two Liberal Democrats and one from the SNP.
Scottish Tory MP Andrew Bowie said there were several “unanswered questions” about the Murrell case and said it was “incumbent on MPs to get to the bottom of it”.
But SNP Westminster leader Dave Doogan, his party’s sole representative on the Scottish Affairs Committee, has argued against any attempt to “weaponise a taxpayer-funded Westminster committee to take pot shots at their political opponents”.
Murrell is due to be sentenced on 23 June.