West Ham manager Allardyce believes he has unearthed a first-class gem in Adrian, signed from Real Betis in the summer.
Allardyce thinks the 26-year-old is vastly underrated simply because he moved on a free transfer.
Allardyce has long admired Finnish stopper Jaaskelainen's abilities, since managing the 38-year-old at Bolton.
Former Bolton boss Allardyce brought the Finland keeper to Upton Park last summer, but conceded he had to identify his long-term successor.
"I would say the goalkeeping jersey is up for grabs, I wouldn't say it's either's now," said Allardyce.
"It was Jussi's most of the season but Adrian's slowly been introduced into the phrase of this fantastic football league we have in this country, by playing in some really big games in the Capital One Cup.
"We played Cardiff and Burnley, and Cheltenham was a hard game first just to get through that.
"Then the way he played against Tottenham, to perform like he did, he deserved an opportunity to go in again at Old Trafford.
"There's not a lot between them in goalkeeping terms at the moment.
"The Juss man can't go on forever, but he's doing an unbelievable job by still playing at this level at his age.
"He's fantastically fit and extremely dedicated, and that's why he's where he is today.
"If I had paid £10million for him everybody would be talking about him.
"But because I've done the job and got him for nowt everyone asks who he is.
"He played every game for Real Betis last year and they finished fifth in La Liga."
Allardyce admitted the toughest part of trying to recruit a defender in the January transfer window will be convincing a rival club to accept a loan deal, not a permanent move.
He said: "We're having to use George McCartney as a cover centre-half if anything happens to James Tomkins or James Collins, so that area we're looking at in January.
"The hard part for us, it's a loan player we're looking for.
"And at this early stage of the transfer window most clubs if they are letting a player go would prefer to sell them rather than loan them.
"But we'll keep going and going and going until we find somebody."
West Ham host Arsenal at the Boleyn Ground on Boxing Day, with Allardyce wary of the growing resolve in Arsene Wenger's side.
Hailing the return of midfielder Matthieu Flamini as adding extra bite, Allardyce said: "They are not conceding from set plays and corners (like) they used to be pretty frail on.
"And they've already scored six headers.
"It's probably why they're up at the top of the league because they've improved in those two areas, and with all the flair they've got as well it's taken them to the top end of the table.
"Perhaps Flamini's been the big difference, going around geeing all the players up.
"When I see him on the pitch he not only leads by example he also leads by the fact that he seems to go around saying 'come on lads'.
"You've got to do both sides of the game, you've got to be just as good as you are out of possession as you are with it, otherwise you're not going to win the league, that's for sure."
- Sports & Recreation
- Real Betis