The 36-year-old defender began his career with the club and returned in 2011 after featuring for Blackburn, Leicester and Preston south of the border.
He made his 100th appearance in the final day 2-0 home win over Motherwell which saw the Saints finish third in the SPL and secure Europa League qualification.
While admitting it was great to go out on a high if he fails to win a new contract, Davidson would love to stay on and help the club build for the future.
He told the Daily Record: "I know I can still play but it's not my decision. It's a shame we got to the last game and there was still no word on a contract.
"I'm not a guy who will muck about so it's frustrating nothing has been said one way or another.
"The ideal scenario is to stay here and combine playing with some coaching. But if nothing is said in the next few days I'll have to look around for another club.
"It was a great day against Motherwell. I gave tickets out for the game to more than 100 Primary 6 and 7s after a coaching session at Dunblane so hopefully we've won a few more fans going forward.
"My family were in the lounge for the game because I've been treating it as my last for the club. It certainly looks that way - there are a few boys who don't know if they will be back next season.
"The last time Saints finished third was not long after I left to go to Blackburn in the nineties. I've told the club I'm keen to stay but if this was my last game it was great to bow out on a high.
"It's pretty amazing to finish third. It boils down to the gaffer, Tommy Wright, and a great bunch of boys here."