The 42-year-old took over from Paul Hartley in April with the aim of keeping the club away from relegation, and he was able to do that with two wins and a draw, which resulted in a 10th-place finish in the Scottish Premiership.
Ross has been selected as a top contender for Dundee, but they have been knocked back following a formal approach.
A statement from St Mirren chairman Gordon Scott said: "We can confirm that Dundee FC submitted a request to discuss their managerial vacancy with Jack Ross.
"We have replied informing them that Jack is under contract and that we won't be giving them permission to speak to him. That is the end of it as far as we are concerned."
Ross took charge of St Mirren in October last year.