The 32-year-old Australian is out of contract with the Perth club in the summer and is keen to leave his native country to try his hand overseas.
It is thought he is a prime target for Exiles after their interest in Melbourne Rebels' Richard Kingi subsided.
A particular attraction is Sheehan's experience, with his three Test caps complementing a number of appearances with Western Force, New South Wales Warratahs and Queensland Reds.
At Western Force he helped steady the ship when fly-half Willie Ripia was injured and, crucially, offered strong leadership in a young back division.
It is hoped that he could do the same at Exiles, who have a number of academy players coming through and are losing Bob Casey, Nick Kennedy and Paul Hodgson in the summer.
With Delon Armitage also set to depart for Toulon, Sheehan's experience would be a welcome addition to a side brimming with talent and potential.
Exiles are also thought to be interested in securing the services of 34-year-old fly-half Stephen Jones, who would link up again with former Scarlets team-mate Steven Shingler.
And, at the moment, it is reported that Irish are the only Premiership side in the race to sign Wales' record cap holder Jones, although there is interest from abroad.
Toby Booth's men are without a game this weekend but return to action on Sunday, March 25 for their St Patrick's Day clash with Leicester Tigers.
Exiles are hopeful of having a near-full strength squad for the encounter, with Ofisa Treviranus expected to be available after returning from Samoa.
The 27-year-old has played just three times for Irish as went home at Christmas for the birth of his child and then saw his return delayed.