The only vacant seats for 2016 are at Marussia - set to compete as Manor Racing this season - with the team yet to confirm either of their two drivers.
Rossi already has links with the team, having raced for them five times last year, including making his Grand Prix debut at Singapore.
"Talks are [happening] every day," Rossi told British F1 broadcaster Sky Sports while attending the 2016 Zoom Auction for Great Ormond Street's Children's Charity.
"I think the deadline really is when cars roll in Barcelona in a couple of weeks. Things sometimes get pushed out in this sport and it is something I am used to.
"Hopefully it doesn't go on too much longer, but I'd say things are looking positive and I hope to get it done very soon."
Manor are expected to make major steps forward in 2016, having made a number of personnel changes and agreed a deal to use the class-leading Mercedes power units.
Alongside Rossi, the team have been linked to Indonesia's Rio Haryanto, who has spent the last few seasons competing in GP2. He won three times and claimed five podiums in the series last year, finishing fourth overall, two places behind Rossi.