The American Federation of Television and Radio Artists called the dispute a "knotty problem" that had no precedent in decades of negotiating.
The three-year contract expires June 30, and the larger Screen Actors Guild is set to resume its negotiations with the studios Wednesday. Both unions and the producers have said they sought to avoid a strike.
"We are trying to think out of the box in order to reach pragmatic results," AFTRA president Roberta Reardon said in the message to its members. "We are prepared to bargain continuously, for as long as it takes, including working straight through the Memorial Day weekend."
The dispute over Internet clips has its roots in the 50-year tradition of actors having the right to consent to the use of clips in noncommercial, nonpromotional situations such as when a TV set is playing inside a movie.
Such permission was rarely requested -- partly because each performer in the clip would be guaranteed at least the day-player minimum wage, now $759.
The studios, represented by the Alliance of Motion Picture and Television Producers, have argued that the process of consent and compensation needs to be streamlined to compete in an Internet age when pirated copies of clips are already freely distributed.
"The existing rules, and the enormous administrative burden created by those rules, prevents the entire industry from developing a lawful clips market," the alliance said in a May 20 news release.
SAG has also highlighted the issue of consent for use of clips as one of the key sticking points in its talks. SAG's contract, covering prime-time TV shows and major studio movie productions, also expires June 30.
On Friday, the Los Angeles Times waded into the debate, urging actors in an editorial to shift "from a self-protective crouch into a more market-oriented stance."
"The goal should be making that market a lucrative one rather than conceding it to outlets that don't pay," it said.
But some actors still had concerns.
Alexandra Leighton, who landed the role of Cherie Hyatt on the upcoming CBS drama "Swingtown," said the worry was about Internet mash-ups, specifically over "I hate to say it, but porn sites."
"There are lots of actors at risk because of stock footage," she said. "I'm not worried about how the studio is going to use it, I'm worried about how Joe over in England is going to use it."
The producers alliance, which declined to comment on Sunday, have proposed to pay a share of revenue to actors in lieu of asking each actor permission, which it said would be unwieldy.
SAG last week warned its members about the problems with giving away their right to consent to the use of their images in clips.
"They could be edited, mashed and morphed into anything, anywhere," it said.