Toulon coach Bernard Laporte outlined his hard-line stance ahead of Thursday's scheduled meeting with Gatland, who has flown to France specifically to discuss the thorny issue of player release.
The Lions open their 10-match tour against the Barbarians in Hong Kong on June 1, the same day as the Top 14 final.
Laporte told L'Equipe: "The international window begins on the weekend of June 9. Everything that happens before that is outside of the window.
"As far as freeing up Gethin, in principle, it's a no.
"If he plays in the final, he will leave the day after, on Sunday June 2, but not before.
"We won't pay a player who has to go and play for the Lions at the most important time of our season.
"There is no specific clause in his contract about it."
Unless Gatland succeeds in softening Laporte's stance, Toulon would throw a significant spanner in the Lions' selection plans by reaching the Top 14 final.
Jenkins is one of the world's leading loose-head props and he would be a strong contender to be in the Lions' starting front row for the three-Test series against Australia.
But Gatland is adamant his whole squad must depart together on May 27, with injury replacements the only permitted late arrivals.
And the Lions coach conceded he may have to overlook players not available to travel that day, however important they may be.
"You don't want to cut your nose off to spite your face in terms of making those players unavailable if they're really, really important for you," Gatland said on Tuesday.
"But I'm also really, really conscious that you want the whole squad together. You don't want people coming out late on a Lions tour - you want them in right from the start.
"If it means us having to make a really, really tough call because someone's not available because they made a decision to go to France, we might have to make that call because it might be the best thing for the squad.
"At the end of the day, it's all about what's best for the team.
"We're not going to make a hard and fast rule, but we are capable of making tough decisions if that's right for the squad."
The Lions coach suggested players may have to take matters into their own hands, as Nathan Hines did in 2009 when he missed Perpignan's Top 14 final so he could go on the tour of South Africa.
"Ultimately it may be up to the players," Gatland said.
"In 2009, Nathan Hines left Perpignan before a final because the Lions meant so much to him as a player.
"He said 'bugger playing in the French club final, I want to be part of a Lions tour'.
"As a management board and coaching staff we will have to discuss that because preparation is so tight.
"If it means players aren't available to get on the plane, it may rule them out."
Jenkins is part of a strong British contingent at Toulon, including Steffon and Delon Armitage, Andrew Sheridan, Simon Shaw and Jonny Wilkinson.
Gatland is also meeting with Bayonne and Perpignan this weekend, who have the likes of Mike Phillips, James Hook and Luke Charteris on their books, although those clubs are less likely to make the Top 14 final.
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