Manager Allardyce's future has been up in the air since the turn of the year, with owners David Sullivan and David Gold keen to make sure that the club's Premier League status was secure before extending the contract of the man who got West Ham promoted last season.
Speculation has mounted in recent weeks that the Hammers have been considering appointing a new boss, such as club icon Paolo Di Canio, at the end of the season, but Allardyce revealed earlier this week that he has agreed the framework of a new deal.
Allardyce will not sign the contract until his team arrive at the points haul he thinks will guarantee West Ham's Barclays Premier League status.
"I always want to know we are safe first (before signing)," Allardyce said on Thursday.
"I suppose I could sign a contract that said if we get relegated then I don't want it, but I won't do that.
"We think we will be safe at 38 points. Then we can push on to 40 if we can.
"The contract negotiations are ongoing at this moment in time. We talk confidentially among ourselves and hopefully we can come to an agreement shortly."
West Ham's presence in the top-flight looked under threat when they hit a mid-season wobble, but recent wins over Stoke and Swansea have propelled the east London club to 14th in the table, six points above the drop zone.
The club have been lifted over the last week by the news they will move to the Olympic Stadium at the start of the 2016-17 season.
Allardyce knows that a bad series of results before then could mean he is not the man in charge by the time the club move to Stratford, however.
"If I can last that long (until the stadium opens) I'd like to be here," the former Bolton manager added.
"The sack is always around the corner, that's how it is these days.
"I've said what I have said (about the contract renewal) and it has caused speculation in a negative way.
"But it has never felt like that to me. It's always been what we said at the start of the season - let's see how we go on and sort out a contract as and when.
"I have not seen any effect on the players."
Three more points against West Brom at home on Saturday would be a big step towards safety given that the Hammers have tricky games against Everton, Manchester City, Liverpool and Manchester United before the end of the campaign.
But Allardyce has already started making plans for next season, including sounding out prospective transfer targets such as Vitesse Arnhem's Ivory Coast striker Wilfried Bony.
He added: "We have sorted out pre-season training, player targets are being put in place because we can't afford not to.
"Whether we get those targets is another matter.
"It's been disappointing that we have not already have clinched safety - that has been due to a lack of concentration at times and a lack of goals."
Allardyce welcomes back Joe Cole, who has been out since March 2 with a hamstring injury, but back-up goalkeeper Raphael Spiegel has been ruled out with a broken finger.
Mark Noble will miss out as he has a trapped nerve in his shoulder. Allardyce is unsure whether the key midfielder will play again this season.
He said: "I don't know how long Noble will be out for. We know very little about his injury so we will take it day by day.
"He is being seen by a specialist and is making good progress at the moment."