Warburton was appointed the Rangers manager in the summer of 2015 and has a contract until June 2018.
The Englishman has done a remarkable job at Ibrox and has guided the Glasgow giants to automatic promotion to the Scottish Premiership.
The former Brentford manager is not going to leave Rangers anytime soon, but he will not stay at Ibrox forever.
Brown has suggested that Weir, who is the Gers’ assistant manager at the moment, would be the perfect man to replace Warburton as the number one.
“If Mark ever has designs on going back to England - which he may do - then I would think that Davie would be the one they would turn to,” the Scotsman told The Scottish Sun.
“Whether Davie wants to go back to England I don’t know, his family is still there. At the minute, knowing David, he won’t be thinking about that - it’ll be how well he can do at Rangers.
"But he’s quite astute, he might know what he wants to do further on. I look at Davie’s career and how every step he has taken he has handled no bother. Nothing ever fazed him.”
Weir is a Rangers legend and is very popular among the Scottish club’s fans. The former Scotland international defender did not achieve much success as a manager at Sheffield United, but he is doing quite well in his current role at Rangers.
Rangers have won the Scottish Championship title and the Scottish Challenge Cup this season.