It was previously reported that "Miners" suggested the coach to extend the contract for two years, but he is waiting for offers from clubs in the English Premier League.
Allardyce is under contract for another 12 months but Moyes only until midnight last night, meaning there is no certainty he will still be in charge at the London Stadium next season.
Although they ended the season with a run of three unbeaten games, prior to that sequence West Ham had achieved just one win in eight, with Moyes offering no clarity on whether he would be allowed to remain in his job next season.
The leading candidates to follow Moyes in the West Ham hotseat are as follows.
Are comments from Rafa Benitez a boost for Everton and West Ham?
Rafa Benitez has been discussing his future as Newcastle United manager after guiding the club to 10th in the Premier League in their first season back in the division. Successful spells at Pacos (again) and Braga led to Fonseca moving overseas for the first time to manage Shakhtar in 2016, since when he has won the domestic double and taken the Ukrainians to the last 16 of the Champions League.
West Ham came out of this battle with the points but both sets of fans know they've got a lot of work to do in the summer to stand any chance of getting to where they think they should be - in Europe.
A number of other names have been linked with the vacancy in recent days, including Ronald Koeman, Carlo Ancelotti and Marcelo Bielsa.