Allardyce's days at Upton Park appear numbered, with West Ham not expected to renew his contract at the end of the Premier League season.
The 60-year-old, who avoided the sack last term, has already conceded he does not know whether he will be in charge next season.
However, Downing wants to see Allardyce lead the team in 2015-16.
"We'd like him to stay. We don't know what the situation is," Downing said following West Ham's 1-1 draw with Stoke City on Saturday.
"It's going to be resolved at the end of the season as he said yesterday I think. The only thing we can do is win games.
"This puts him in a stronger position to keep his job. Obviously he is disappointed today because wins help. Losing games and drawing games probably won’t help him.
"Like I said, we are disappointed today. Whatever will be will be at the end of the season. I'm sure he’ll want to get it sorted as soon as he can."
West Ham are ninth in the standings, with 43 points from 32 matches.
