The Hammers will move out of Chadwell Heath to start training at their academy base of Rush Green, as they look to curb their injury problems.
West Ham have had a series of players ruled out this term, and Allardyce says that Bilic is suffering from similar problems to him.
"Yes, he’s not wrong. I think he’s right - if they can get to a new training ground," Allardyce said.
"It’s certainly something I suffered with. We started falling into the same injury problems as Slaven has.
"This time last year we were probably fourth in the league, but because of those injuries we had an indifferent start to the season.
"I think it’s a good move if he can get on to a new training ground."
Bilic has complained about the standard of the pitches at Chadwell Heath, which he feels has contributed towards West Ham’s injury crisis.
The Hammers have made an impressive start to this campaign, but keeping their standards up with so many players sidelined will be a tough ask.
Dimitri Payet, Manuel Lanzini, Victor Moses, Diafra Sakho and Winston Reid are all set for lengthy spells on the treatment table, leaving Bilic’s options looking short.
West Ham have failed to win any of their last five matches, causing them to fall to seventh place in the Premier League table.
The Hammers’ next fixture comes against Stoke City on Saturday.