A source close to the player told the Telegragh that the newly promoted English League One outfit recently had a £50,000 offer knocked back by chairman Douglas Rae.
Midfielder Tidser, 23, last night admitted this was indeed the case and that Ton owner Mr Rae had informed him of the South Yorkshire side's approach.
He revealed: "I've spoken to my agent and he says there was contact and a bid made by Rotherham.
"He deals with that side of things. I haven't had any personal dealings with Rotherham directly.
"I have spoken with the chairman here at Morton myself and he mentioned a figure which he thinks is acceptable. Any interested clubs would need to meet that valuation.
"The chairman owns the club; he pays the wages, so he's entitled to do what he wants.
"I have a year of my contract left to run, so he can either maybe accept a deal from someone and make some money or I can go somewhere for free next year."
Tidser has spent the last three-and-a-half seasons with the Greenock club, having joined on a free transfer as a teenager in January 2010. In that time the former Celtic and Ostersunds FK player has progressed markedly and caught the eye with a string of superb outings as Ton challenged for the First Division title last term.
He scored 13 goals and laid on 17 assists in all competitions despite missing the last four league games and was nominated for the PFA Scotland First Division Player of the Year.
Tidser feels he is now ready to test himself at a higher level and would be keen to move to a club in England, but also insists he is not agitating for a move away from Morton.
He added: "Down south is where you want to go, the SPL or down south, most players will tell you that. You want to challenge yourself.
"Uncertainty in your future can be frustrating because you like to know what's happening and you like to get your mind-set right for the start of the season."But, as I say, it's out of my hands. It's down to the club and whoever else is dealing with any offers.
"I've played a lot of games for Morton; I've been here over three years now, and, if I'm still a Morton player next season, it wouldn't be a problem. I'll knuckle down and try and replicate what I did last season.
"When I signed the three-year contract, my intention was to honour that. You do want to test yourself at a higher level, but there is no problem staying at Morton.
"It's a good club and I've had a great time here so far, and hopefully we can try and do as well as we did last year."
On Rotherham, who play Ton in a pre-season friendly at Cappielow on Tuesday 9 July, he said: "They got promoted to League One last season. I think they're a decent club with a decent stadium but I don't know much about them, to be honest.
"Obviously they'd need to have a bid accepted and I'd then need to talk terms and come to an agreement, so I wouldn't like to say too much on one club as it can always backfire."