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The Robins manager wants Williams to strengthen his leaky defence that has conceded 22 goals in 14 games so far this season.
But Tinnion, who has offered thirty-three-year-old Adie Williams a two-and-a-half year deal, insists that he will not wait around forever for the Royals skipper to decide where his future lies.
"We've made a good offer and it is now up to Adie. There's no deadline at the moment but we won't be waiting forever.
"The defence is one area I want to strengthen and we need to know sooner rather than later if Adie doesn't want to come so we can move on," said Tinnion, speaking to Bristol City's official website.
It seems that Williams may decide to make the move down the M4, enticed by the contract on offer, bringing to an end a Royals career that has spanned more than four hundred games during two spells at the club.
Having been transfer listed during the international break at the start of the month, Williams has not featured in Reading's last three games due to injury.
Given the superb form of Senegalese centre-back Ibrahima Sonko it seems unlikely that the centre-back will play another game for the club, even if he continues to stall over a move away from Berkshire.
Williams was present at the Madejski Stadium last Saturday to watch the Royals thump Crewe 4-0.