"The board have to sit down and tell a new manager ... you've got a three-year deal and within that time we expect improvement in the Premier League table and a trophy.
"If you've not done that within that time frame, you're off," former Welsh international goalkeeper Southall, who made 747 appearances for Everton during a 17-year stint at the club, told British media.
Former Netherlands defender Frank de Boer is favourite to replace Martinez, according to several British bookmakers, after he left Ajax Amsterdam on Thursday. Ronald Koeman has also been linked with the job, although he said on Friday that he expects to stay at Southampton.
"Ideally what we need is a mix of David Moyes' defensive solidity and Roberto Martinez's attacking flair," said Southall. "A Jose Mourinho (type) team would fit the bill perfectly."
Sacked by Chelsea in December, Portuguese Mourinho is tipped for a move to Manchester United if they part company with Dutch manager Louis van Gaal this summer.
Everton reached the semi-finals of both domestic cups this season, but the team's failure to win nine of their last 10 league games cost Martinez his job.
The Spaniard also lost the fans' backing, with some staying inside Goodison Park after last month's win over Bournemouth to voice their discontent over his three-year tenure.
(Reporting by Shravanth Vijayakumar in Bengaluru; Editing by Catherine Evans)
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