The experienced pair will reach the end of their current deals on June 30 and remain training with PNE.
For them to play in the games beyond the end of June, they would have to agree some kind of extension by next Tuesday.
An extension could be to see them through to early August to cover the play-offs should PNE reach them.
Or deals might be struck for longer stays beyond the end of this season.
Next Tuesday is when clubs have to submit their retained lists to the EFL.
In previous years that had to be done in May, however this season running late has seen clubs given longer. Club skipper Clarke and midfielder Gallagher are the two outstanding contract issues for PNE.
Goalkeepers Declan Rudd and Mathew Hudson signed new deals during the lockdown – their previous contracts ran until June 30.
Michael Crowe and Connor Simpson have left the club, with their contracts not being renewed.
Neither were required to report back for training at the end of May.
Gallagher, 35, and Clarke, 32, both have long service at North End and are a big part of the dressing room.
This season Gallagher has started 18 Championship games and come off the bench in eight more, scoring six goals.
He has made 272 appearances for the club since November 2013, initially on loan from Leicester and then as a permanent signing.
In 2007, Gallagher also played 19 times for PNE when on loan from Blackburn. This season, he has been working with the Under-16s in the academy while working towards his coaching badges.
Defender Clarke has pulled on a Preston shirt 242 times since joining in June 2013 on a Bosman from Huddersfield.
Game time has been more limited for him this season, with him to date starting seven Championship games and making three appearances from the bench.
North End have two games in June – at Luton Town on Saturday and against Cardiff City at Deepdale a week later – before Gallagher and Clarke’s current deals end.
Other Championship clubs are in similar positions with players who are coming to the end of their contracts.
In some cases, players have opted not to sign short-term deals to cover the remainder of the season.
That is presumably because they want to avoid injury which would hinder a move elsewhere during the summer.