Wigan chairman Dave Whelan says he wants Martinez, who met Liverpool's principal owner, John Henry, in Miami, to make a decision on his future by Thursday.
Former Barcelona and Bayern Munich coach Louis van Gaal has been heavily linked in the British media with the job of replacing the fired Kenny Dalglish, while Swansea denied on Monday the club had received a second approach from Liverpool for highly rated manager Brendan Rodgers.
If Liverpool's American owner, Fenway Sports Group, move for Martinez, he will be the club's second Spanish manager in the past decade, after Rafael Benitez.
Roberto Martinez to resume talks with Liverpool
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