It is understood Wigan will hold a press conference this week in which Martinez is expected to announce whether or not he will stay at the club.
The 38-year-old met Liverpool's owners in Miami on Thursday to discuss the managerial vacancy and was not put off by the huge challenge that would face any new manager.
However, it is understood Martinez has not made up his mind completely whether to accept that challenge and has given himself until Tuesday to do so.
Wigan chairman Dave Whelan claimed on Friday the Spaniard had received an offer from Liverpool's principal owner John Henry after their meeting in Miami on Thursday.
Doubts emerged over Whelan's interpretation of the situation, with the Reds refusing to comment amid the general belief no formal offer had been made.