The former West Ham, Derby, Blackburn and Dundee United player did not make enough appearances to merit a league medal but leaves with a Scottish Cup winners' medal, having come on as a late substitute during Saturday's 1-0 Hampden win against Falkirk.
Talking after the cup triumph, Weir said: "I've always said how much I enjoy playing at Rangers and. if the manager wants me to stay, I'd be delighted.
"We've had a successful season, I've been a part of that, I've enjoyed it and hopefully I'll be here next year."
Manager Smith added on Monday: "Davie's 39 and we have an agreement that we sit down and talk at the end of each season.
"I had a few words with him just before the cup final. He's keen to continue and there's a willingness on Rangers' part to have him for another season.
"Both parties will sit down over the next few days and see what can happen."
Weir began the season as captain, with Barry Ferguson missing through injury, and he took over the leadership duties for the close of the campaign once his midfield colleague had been stripped of the honour for his high-profile misdemeanours while on Scotland duty.
Smith also reiterated his desire to stay at the helm for at least another season.
"It's a natural a thing when you get over the age of 60 people start asking when you'll retire," said Smith.
"Normally in football you don't have the choice. The success that we've achieved this season is a motivation for me to continue.
"If we had lost out in the league for another season or lost out in the Scottish Cup, I might not have had the option. But if I had, disappointment comes into it and you may have looked upon it in a different manner.
"But winning again and getting your hands on the championship trophy and going into the Champions League for another year is more than enough motivation for me to carry on."
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