The 27-year-old will now see if he can attract any offers to go on loan as a No1 keeper having hardly featured since his summer move to west London.
A temporary exit to earn more gametime, however, will only be accepted should the Bees be satisfied by the offer made by an interested bidder.
Manager Thomas Frank is preparing for an FA Cup match with West Ham on Saturday where Strakosha is expected to make his second start for the club, in place of long-term No1 David Raya.
Strakosha is not understood to be desperate to leave and is attempting to force his way back into the Brentford team after signing a four-year deal over the summer.
Brentford reportedly beat Manchester United to sign the former Lazio goalkeeper on a free transfer over the summer.