The 20-year-old is understood to have left the Blues' pre-season training camp in the US, jetting back to England to facilitate a move to the Hammers.
Having reportedly agreed personal terms with the East London club, Broja's length of contract is still be negotiated between both parties.
"This Armando Broja story is developing rapidly. We know he's left Chelsea's tour of America to thrash out personal terms with West Ham," talkSPORT's Alex Crook said.
"I told you this morning that Chelsea want a loan deal, but Broja and his agent are really pushing for a permanent transfer. West Ham want that as well.
"What we will probably see is a permanent transfer, of around £30m, and maybe Chelsea might insist in a buy back option, like we've seen with Man City letting some of their youngsters go out on permanent deals."
Broja is keen for regular first-team football, and has made just one appearance for Chelsea since arriving in 2009. He has, however, impressed at Vitesse and Southampton on-loan.
The Albania international netted six goals in 32 games for Ralph Hasenhuttl's side during the 2021/22 season.
Everton were understood to be keen on the striker, but Broja had his heart set on living in London.
The Albanian ace has already made his mark in the capital, and is remembered by Spurs fans for having celebrated against them.
He has starred with the Saints and scored against them in their Premier League clash to draw the sides level last season, after Tottenham went ahead through an own goal.
Broja reacted quickly to fire the ball into the back of the net, past Hugo Lloris, and the 20-year-old proceeded to celebrate wildly infront of the home supporters.
The Slough-born forward had started his career with Tottenham as a child, joining their academy from the age of eight.
Should Broja sign for David Moyes' side, he will be hoping to have another fine campaign, and help guide the Hammers to European football.
West Ham are also understood to be keen on a left-back, and remain in the hunt for Sassuolo striker Gianluca Scammaca, but do face a fight for his signature with Paris Saint-Germain also interested.