The Easter Road boss has begun talks with the 35-year-old Finn, who was at npower League One side Oldham last season, in the hope of adding him to his squad for Saturday's Clydesdale Bank Premier League clash against St Mirren.
Fenlon, who has lost midfielder Lewis Stevenson to a broken toe, said: "We have started talking and will see how we progress, we may not have the deal done by the weekend but we will keep trying."
Fenlon has a list of targets as he aims to put his own stamp on proceedings at the club.
And the Irishman will not rule out the arrival of a few more new faces before the end of the transfer window.
"We still feel we are a little short in a few areas and we are still looking to get new faces in, we are not sitting on our hands and are very hopeful," the former Bohemians boss said.
Stevenson suffered a toe injury in last weekend's 1-1 derby draw with Hearts, coming off early in the second half.
"He damaged his foot in the derby last week and will be out for around two weeks. Other than that though, everyone has come through okay," Fenlon said.
The likes of Northern Ireland defender James McPake and Scotland Under-21 pair Paul Hanlon and David Wotherspoon reported back fit from international duty.
Looking ahead to St Mirren, Fenlon said: "It's difficult as you only have a couple of days to prepare for the game, however I suppose we have to be pleased that we have five players playing international football in the midweek.
"From a manager's point of view it can be a bit of a nuisance, but it is what it is and is no excuse for not being ready for the game on Saturday."