The 31 year old, who has won 88 caps for Romania, could be on wages up to ?40,000 a week, for three years and the Hammers could snap him up on a free transfer.
Rat signed for Shaktar Donesk and won six league titles with Shaktar from 2003 on-wards and has made over 170 league appearances for the side and scoring 6 goals.
The earliest the deal can take place would be theend of June when his Shaktar Donesk contractexpires.
Rat would bring more experience to West Ham'sinternational defence.
Defenders; James Collins, George McCartney,Winston Reid, Joey O'Brien, Guy Demel and and Emmanuel Pogatez round up a total of 184 international caps and if Rat does join the East-London club he will become the most experienced player at the club.
The last Romanian to play of West Ham was striker Florin Raducioiu who played eleven games for West Ham during the 1996-97 season.
Allardyce's West Ham has been hit with injuries withleft-back George McCartney being out for most of the2012-13 season.
"To improve we just need to add better players, it's a simple as that," Allardyce said last week.
"If we get better players, then we improve. If we can do that, thenwe get better and we become an established Premier League side."