Jaaskelainen has held the number one jersey at Upton Park since arriving on a free transfer from relegated Bolton in the summer but Allardyce is concerned with a recent drop in his performances.
The West Ham boss believes a lack of competition is to blame, reserve stopper Stephen Henderson being the only direct rival, and he is now pondering a move for Robinson before the transfer deadline.
The 33-year-old worked alongside Allardyce during his two-year spell as Rovers manager between 2008 and 2010 but has found himself warming the bench of late with new boss Michael Appleton preferring youngster Jake Kean since he took over from Henning Berg.
Its believed Robinson is open to joining the Hammers and will swap roles with 24-year-old Henderson, who spent three-months on loan at Ipswich earlier this season, in a part-exchange deal.
Personal terms are have yet to be agreed but that is expected to be a formality with the former Leeds and Tottenham keeper ready to move to return to London despite privately expressing a preference to stay in the north of England.