The hit TV period drama ended on Christmas Day 2015 but has since sparked endless speculation about whether it will return with a film.
Carter, who played butler Carson in the show, told Good Morning Britain that “nobody’s seen the script”.
But he said: “We’ve been asked to keep ourselves available at dates in the future...”
He added of the script: “There’s a place in the West Country called Julian Fellowes’ head and it’s in there somewhere. I’m sure it will emerge.”
He told Good Morning Britain host Susanna Reid: “I think there’s a willingness there to do a film but until there’s a script there’s no film.”
Carter said he was meeting the cast of the ITV drama on Thursday night.
“I’m having dinner with a few of them tonight. And I’ve just gone dry January ... We’ll all be gossiping about this very topic probably.”
His comments come after Joanne Froggatt, who played maid Anna Smith, said she had heard rumours that Downton’s creator had already penned the script.
“We’ve all heard this rumour - the cast - that the script has been written,” she said.
“There’s a hard copy somewhere but none of us have seen it or know what’s in it. So we’ll all keep our fingers crossed that it all comes together. We’re all willing it to happen. I’d absolutely jump at the chance to star in it if I’m asked. I’ll squeeze it in.”
Fellowes has also spoken of the possibility of a film, saying the difficulty was getting cast members - such as Hugh Bonneville and Michelle Dockery - free.
“The difficulty is rounding up the actors. One is in America making a new series, another is in a play,” he has said.
“My lips are sealed! It’s very unlikely there won’t be a film because there is an audience out there for it.”
Michael Fox - who portrayed footman Andy Parker in the series - has suggested that Dame Maggie Smith could reprise her role as the acerbic Dowager Countess in the drama.
“She elevates it to something else, because she is such a legend and a national treasure. The film without her would still be good with all the other actors, but it puts a stamp of quality with her in it,” he said.
The final episode of Downton Abbey was the most-watched TV show on Christmas Day in 2015.