His exit seems increasingly inevitable after a poor finish to the season.
The Toffees are currently 12th in the Premier League table but have won just one of their last 10 league games.
Supporters' groups are planning a protest at St George's Hall in Liverpool, where players and officials will gather on Thursday evening.
The only potential sticking point is that Martinez still has three years left on a lucrative contract, worth a reported £4m a year, that he signed in the summer of 2014.
Everton finish their season at home to relegated Norwich City on Sunday.
The agent of Dutch coach Frank de Boer, who has just stepped down at Ajax, has stated he would be interested in the Everton job if it became available.
Southampton's Ronald Koeman and Manchester City's Manuel Pellegrini have also been linked with the club.
Everton's board, led by new major shareholder Farhad Moshiri and chairman Bill Kenwright, have remained silent on Martinez's future.
It had been expected they would wait until the end of the season before conducting a review but anger among Everton fans means a decision on Martinez's future could now be made earlier.
Everton have produced three dismal away performances in succession in the Premier League, losing 4-0 at Liverpool, 3-1 at newly-crowned champions Leicester City and 3-0 at Sunderland on Wednesday.
They also lost the FA Cup semi-final to Manchester United.
In addition, Martinez has presided over the worst home record in Everton's history this season, losing eight of their 18 games so far.
They have won only five league matches at Goodison Park, with two of those coming against relegated Aston Villa and Newcastle United..
Martinez was appointed Everton manager in summer 2013, succeeding David Moyes after he left for Manchester United.
The 42-year-old Spaniard guided Everton to fifth place with 72 points in his first campaign - a club record in the Premier League era - but the Toffees managed just 47 the following season, finishing 11th.
Everton boss Roberto Martinez will face fans' protests at the club's end-of-season awards dinner - unless he is sacked before Thursday's event.