The midfielder is without a club following his release by Fulham at the start of July, but has been offered an opportunity with the Lancashire club to stay in England on trial.
Sheffield United are also reportedly interested in signing the player, but the lure of playing for his third Premier League club could prove crucial.
The 27-year-old initially impressed at Bolton in 2007, playing 21 times for the Trotters and scoring four goals.
However after moving to Fulham on a free transfer in 2008 he made just one appearance and was subsequently released from Craven Cottage.
Blackburn have already played Teymourian in one pre-season match against Southport, but are set to give him another chance against either Preston or Huddersfield tomorrow, with coach Steve Kean eager to see if he can impress further.
"He has trained very well and I think he had a little bit of a tough time when he was at Fulham and never played too many games," Kean told the Lancashire Telegraph.
"But he certainly had a good impact on the sessions and Saturday in the game.
"But he needs match practice to get back into the swing of things."