A Jaguar spokesman said on Monday that the 20-year-old, who had a strong first test with the team in Spain last week, had been recalled for 'further evaluation' at Barcelona's Circuit de Catalunya on Tuesday.
Jaguar could not say how long Klien, backed by Austrian energy drink sponsor Red Bull, would be testing for.
However team boss Tony Purnell has set an early-December target for confirming the line-up and sources hinted that a deal with the Austrian now looked imminent.
Klien, runner-up in German Formula Three this year, impressed Ford-owned Jaguar with his speed and attitude at the Valencia test and the team would also welcome the injection of cash that he can provide.
The Austrian lapped just a tenth of a second slower than Webber, albeit on different tyres and fuel levels.
"We haven't finalised anything yet," said spokesman Nav Sidhu. "But he did a fantastic job. He certainly ticked all the right boxes."
FRESH TALENT
Klien's confirmation would be a serious blow for Briton Justin Wilson, who ended the 2003
Wilson's contract runs out at the end of the year and the 26-year-old, who raised $1.6 million by selling shares in himself to get a drive with Minardi at the start of the year, will struggle to find another opening.
New rules for 2004 to encourage fresh talent allow mid-table teams like Jaguar to run third cars in Friday practice but only with drivers who have competed in fewer than six races in the last two years.
That would rule Wilson out of any testing role.
The Jaguar drive is the most sought-after vacancy remaining for 2004, with only Jordan and Minardi yet to announce their line-ups.
Austrian Alexander Wurz had been a frontrunner for the position until he ruled himself out last week by announcing that he was staying at McLaren as test and reserve driver.
Dutchman Jos Verstappen was another aspirant as was Germany's Nick Heidfeld, who has signed an option with Jordan and begins testing with that Ford-powered team in Barcelona on Wednesday.