Sunderland have signed experienced full-back Anthony Reveillere on a deal until the end of the season.
Norwich have recalled striker Carlton Morris from his loan spell at Oxford.
Allardyce rang co-chairman David Sullivan on Thursday after reports emerged suggesting he faced the sack after the team's final game of the season at Manchester City on Sunday.
Hammers fans have regularly vented their frustration over the club's style of football under Allardyce and Sullivan apologised for the campaign at the club's end-of-season awards dinner on Tuesday night.
Allardyce is due to meet Sullivan next week but the Hammers boss does not believe his position will be on the agenda.
"I rang him up and spoke to him about what's been planted around and under those circumstances we just talked about getting together and having our meeting next week," Allardyce said.
"The only thing I can say is I spoke to David Sullivan yesterday about us arranging our usual end-of-the-season meeting to go through pre-season and go through budgets and what will happen next season.
"I had no indication that my position is under threat. That's all I know, he's the boss and that's the bottom line for me."
Allardyce added: "People in the game have to write speculation because they have to sell newspapers.
"Whether that speculation becomes a reality we don't know but I will find out next week if it is."
West Ham are 12th in the Barclays Premier League table and have secured their top-flight status for a second consecutive season.