Allardyce was summoned to a meeting at West Ham co-owner David Sullivan's house in Essex on Tuesday where it is understood he was asked to put forward his case to remain in charge of the club next season.
The meeting, which lasted approximately two hours, was also attended by the club's chief executive Karren Brady and co-owner David Gold.
Allardyce's views were understood to have been noted by the West Ham owners, who will first convene a board meeting before deciding on whether he will remain in charge next season.
That board meeting is expected to take place in the next seven to 10 days and the club are understood to have set a deadline of May 23 to decide on Allardyce's future.
Reports had suggested the 59-year-old was facing the sack after a difficult season, with the Hammers leaving it until the closing weeks to secure their Premier League status.
Despite criticism from a section of the fans, Allardyce was satisfied with the club's 13th place finish, but it remains to be seen if West Ham's board also share this view.
Allardyce took charge of the Hammers in 2011 and immediately steered them back to the top-flight, but some supporters have never warmed to his playing style and jeered after their home win over 10-man Hull City in March.
The former Bolton boss responded defiantly to questions at the weekend, pointing to his long managerial record as proof of his success.
"If you are a manager, this is the highest level in the world," he said.
"I have competed at this level for many, many years. There's only Arsene Wenger has competed at this level longer than me.
"I know how to manage in this league and I know how to turn teams into better teams than when you first take them over, so there you go."