The 37-year-old played under the guidance of the Hammers boss at Bolton for eight years as the pair helped establish the Trotters in the Premier League.
Jaaskelainen revealed that it was an easy decision to join West Ham as soon as he was aware of the interest from his former manager.
"I'm absolutely delighted to be here under Big Sam again, having worked with him before for a few seasons,” the goalkeeper told the club’s official website.
"It happened very quickly because when my agent said that West Ham were interested I told him to sort it straight away because I know how Big Sam works, how well he looks after the players and how good his backroom staff are. They were the factors behind why I wanted to come here.
"He brings certain routines and that makes it a lot easier."
The Finland international is eager to get the pre-season with his new team-mates underway, with securing the No.1 jersey his main focus.
Jaaskelainen continued: "At the start of pre-season, all the goalkeepers are in the same place, but obviously I want to go and get that No.1 place and keep that consistent level I have done over the years.
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"The main focus is playing for West Ham and to work hard for this football club and if I get to see London then that's good, but it's not my main focus.
"Pre-season is a little bit different this time because I didn't play any games this year [for Bolton], so all off-season I've been looking forward to coming back and training again.
"Normally in the off-season you're tired from playing all season, but this year is a little different so I'm pleased that we've started a week earlier than the other teams.
"We have seven or eight weeks before the season starts so we should be well ready when it does."
Jaaskelainen is hopeful he and Allardyce can repeat the feat from their time together at Bolton and secure West Ham's position in the Premier League following their promotion from the Championship.
"Obviously the aim is to stay up in our first season," he added. "It's all about staying in the Premier League and bringing in more money and improve the facilities at the football club.
"Hopefully I can play here for the next two or three years."