The highly-rated boss led Wigan to a memorable FA Cup final victory earlier this month but was unable to save Latics from relegation and has been strongly linked with vacant managerial positions at Everton and Stoke.
Wigan chairman Dave Whelan has made it clear that he is hopeful Martinez can be persuaded to stay and try to mastermind the club's return to the Premier League.
Whelan said on Wednesday afternoon: "I've just had a very constructive meeting with Roberto.
"He was very honest, as always, and put his cards on the table.
Not a straightforward decision
"He's got certain things to think about, which I respect. It's not a straightforward yes or no.
"He wants 24 hours to talk to his wife and we'll meet again tomorrow.
"I respect him enormously and, if he says he wants to leave, I won't refuse, but I hope he says he wants to stay here for another year, two years, three years."
Asked if he was optimistic Martinez would decide to remain at Wigan, Whelan added: "With Roberto, I'm always hopeful.
"He likes a challenge and it's a big challenge next year. The Championship is a challenge and we are in Europe as well. We'll need to add to the squad and there is a lot to do.
"He wants 24 hours, which I think is right, and we'll have his decision tomorrow. Whatever decision he makes, I will respect it."
A popular figure at Wigan as a player between 1995 and 2001, Martinez became manager in the summer of 2009, after a successful stint at Swansea.
Wigan finished 16th in Martinez's first two seasons and then 15th last term after a brilliant run of form towards the end of the season, but they finished three points adrift of safety this time, a 4-1 loss at Arsenal last week consigning them to the drop.