The 38-year-old will become a free agent once his deal with the Hammers runs out at the end of the season.
"My contract is until the end of the season and [earning a new contract is] what my aim is at the moment," he told the club's official website.
"But I want to go on as long as I can. The goalkeeping coach and backroom staff are fantastic and look after me pretty well.
"They know my needs and give me one or two extra days off now and then. Experience as a goalkeeper is important and the main aim is to be consistent."
Jaaskelainen joined West Ham in 2012.