The 40-year-old Dubliner has carved out an impressive reputation during spells with Shelbourne, Bohemians and with the Republic of Ireland's under-23 side. He has won the Irish league title five times in seven seasons as a manager.
Fenlon is understood to have met United chairman Stephen Thompson in London yesterday and was interviewed for the job vacated by Craig Levein last month.
Thompson, who also has Raith Rovers manager John McGlynn and ex-United player Michael O'Neill on his shortlist, is leaving no stone unturned in his efforts to find the right man to lead United forward.
And that man could be Fenlon, who guided Shelbourne to back-to-back league titles in 2003-2004 and into the Champions League.
Fenlon masterminded a victory over Croatian giants Hadjuk Split in the qualifying rounds before the Irish assault on the competition was ended by Deportivo La Coruna in the stage before the groups.
The Irishman won the Irish title again in 2006 and was poached by Derry City when Stephen Kenny departed to take over at Dunfermline but left after only five months.
Fenlon was then employed by the FAI to oversee the nation's under-23 side and combines this role with managing Bohemians.
It is his work at Dalymount Park, where he led the Bohs to the title by a massive 19 points last season, which has brought him to the attention of United's board.
Thompson, who spoke with John Collins but won't be moving for him, is also an admirer of St Johnstone's Derek McInnes but the issue of compensation means this is unlikely to happen.
Former United manager Paul Sturrock can also be ruled out.
Meanwhile, current caretaker Peter Houston is continuing in his role under the assumption he will take charge of this weekend's cup tie against Partick Thistle.
United hope to have their new man in place by the weekend and even if he is Houston is determined to be able to furnish him with as much information as possible. He said: "I'm going to the Partick game on Monday night and I've also got three DVDs of their most recent matches.
"We're just preparing for Saturday as normal and as things stand I'm going ahead assuming I will be in charge.
"There is every chance we will have a manager in place by then but I know the chairman is being very thorough.
"The last thing you want to do is rush into anything, especially because of the results over the last two games.
"It's vital to take your time and get the right person in these situations.