Rat is available on a free transfer after seeing out his contract at Shakhtar Donetsk and Allardyce has targeted the 31-year-old to help fill the club's problematic left-back spot.
Talks are thought to be at an advanced stage and Allardyce is hopeful of making Rat his first signing of the summer.
"I think at the moment it is talking terms and conditions and if we get through that then hopefully it's a chance of a medical and then see where we go from there," Allardyce told Sky Sports News.
Allardyce believes Rat will prove a valuable addition to his squad next season.
"He has played Champions League this year, if everything goes to plan we get a player that has played at the highest level for his country and for Shaktar Donetsk at the moment," added Allardyce.
"His experience and talents from abroad bringing them into the Premier League would be a great asset for us if we can sort it out."