The Latics owner has a list of 40 applicants who have declared an interest in replacing the Spaniard, who has left the DW Stadium to take over at Everton.
Martinez's four years in charge at Wigan ended after the club won their first major trophy when they shocked Manchester City to take the FA Cup, only to be relegated from the Premier League just three days later.
"On Sunday or Monday I intend to start interviews," Whelan told Sky Sports News. "I'm not going to mess about for too long, I expect to be completed within four or five days.
"We have something like 40 people who have applied for the job, and amongst that 40 people there are some wonderful names and some great managers. I hope I can choose the right one.
"It's a great list, I've got to be lucky and choose the right person, because the right person is on that list."
Whelan also declared his intention to hold on to the club's prize assets as they attempt to get out of the Championship at the first time of asking, most notably highly-rated winger Callum McManaman.
"Callum has got the makings of a quality star footballer, and I want him to stay with Wigan for another year or maybe two. However, we just sit and wait. We want to keep every player we have got because we are going to fight like hell to get back in the Premier League."