The LOST IN TRANSLATION star has met Gyllenhaal a few times, and friends say that the pair are perfectly matched.
Now she has asked her representatives to organise her a date with the actor - who recently ended his long-term relationship with actress KIRSTEN DUNST.
A pal says, "He's a little younger than her usual type, but they really clicked.
"They both grew up in dynamic and creative families and Jake has an old head on young shoulders, just like her."
Johansson, who is rumoured to have had a fling with Benicio Del Toro at last year's Oscars, makes no secret of the fact that she is looking for love.
She recently said, "I'm a girl in the prime of my life and I'd like to celebrate that with someone.
"The thing is, I'd definitely like to be in a relationship with the right person. That's why I'm looking."08/08/2004 10:30