"The internet is just rumours but for sure there are discussions ongoing, Ross has said, with the team," German driver Rosberg, winner in Monaco and Britain this year, told reporters at the Japanese Grand Prix.
"For me, it would be great if he stays because I get on well with him and I think he's a great team principal.
"At the same time I have confidence in the team leadership's ability to make sure we have a solid future whatever his decision is because we have a lot of great strength in the team now as it is."
Reports emerging in Germany after last weekend's Korean Grand Prix suggested Brawn had informed the team's non-executive chairman Niki Lauda that he would be leaving at the end of the season.
Some have speculated that the Briton could end up at Honda, whose team he led before it became the title-winning Brawn GP and then Mercedes, when they return as engine partners to McLaren in 2015. McLaren sources have dismissed such talk.
Lauda subsequently told German media that he was in negotiations with Brawn and no decision had been taken.
"My goal is clearly to retain him, but he will only make the decision at the end of the year," the Austrian was quoted as saying.
Brawn's future has been the subject of speculation all year since the team recruited Paddy Lowe from McLaren as executive director. They also brought in Toto Wolff from Williams as Mercedes motorsport head.
Rosberg said Brawn, the master tactician behind Michael Schumacher's seven wins with Benetton in the 1990s and Ferrari in the 2000s, was highly respected within the team and an important part of it.
Hamilton, who moved to Mercedes from McLaren at the end of last season after being convinced by Lauda and Brawn, agreed.
"I don't know what his plans are for the future, but of course I'd like him to stay," the 2008 world champion, a winner for his new team in Hungary last July, told reporters.
"I grew up watching him on TV, so I had no idea what his personality was like, but he's been just as I thought of him watching TV.
"He's a very easy going guy, very approachable, and is very much responsible for where the team is today and the atmosphere he has helped create."
(Reporting by Alan Baldwin, editing by Patrick Johnston)
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