Mr Mallon said businesswoman Christine Bell and councillor Len Junier had criticised him and his fellow councillors for selling land at Acklam Hall for development.
They referred to it as "dodgy" on Twitter and at a council meeting.
Mayor Mallon said they now had to provide evidence of the claims.
The independent mayor said: "You have two people here who claim the sale of Acklam Hall was dodgy.
"What those people have got to do now is produce the evidence of malpractice, corruption, or criminality and I will give you a cast iron guarantee they will not be able to produce one shred of evidence.
"Now they've actually got to put up or shut up."
Mr Junier (Independent), who has asked the Secretary of State to investigate the sale of the land, said: "This was all about me having a duty to ask questions and raise concerns wherever they exist.
"All I want to know is did the taxpayers get the best deal possible for that land?"
The hall was valued at about Â£1m some years ago but Middlesbrough Council has refused to say how much it was sold for.
Critics said the 32 acres of land was worth more more than Â£20m.
Ms Bell told the BBC she had raised a matter of concern which had not yet been resolved and now awaits the outcome with interest.

The Plane Stupid demonstration on 13 July led to delays and the cancellation of 22 flights.
The activists, who were protesting against plans to expand the airport, allegedly cut a hole in a fence and chained themselves to railings.
Supporters chanted "No ifs, no buts, no third runway!" outside the court.
Sheila Menon, one of the accused, read a statement outside Uxbridge Magistrates' Court. She said the pollution and noise caused by air traffic at Heathrow was already blighting the area.
She said the proposed expansion - recommended by the Airports Commission in its final report last month - was driven by "wealthy frequent flyers" rather than ordinary holidaymakers.
Plane Stupid's actions were "reasonable, justifiable and necessary", said Ms Menon.
The activists were charged with aggravated trespass and entering a restricted area of the airport without permission.
They were bailed on condition that they do not enter Heathrow or its perimeter, with a trial expected to begin on 18 January.

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Gabbiadini scored twice to bring Saints level at 2-2, but had what would have been the opener incorrectly ruled out.
"Gabbiadini's first was a good goal, it should have stood. He scored three good goals," Puel said.
"It's a lot of disappointment. We deserved better," the Frenchman added.
"I would like the video for the future for this situation, but at the moment we are without. Sometimes there is a bad decision against us, it is important to accept this.
"The memories are always for the winner. It's football and congratulations for Manchester. I'm proud of my players because we played very well."
Zlatan Ibrahimovic collected the 32nd trophy of his career after heading an 87th-minute winner at Wembley on Sunday, having earlier given United the lead with a brilliant 19th-minute free-kick.
Jesse Lingard put United 2-0 in front before Gabbiadini struck twice - his fourth and fifth goals in three games since joining from Napoli for £14m.
"Puel has reason to be really sad and disappointed because he did fantastic work and his team deserved extra time,'' United manager Jose Mourinho said.
"To lose in minute 87 with very little time to try to react, as they did in the first half, is a little bit unfair on them."
Goalline technology - to determine whether the ball has crossed the line - was brought in to the Premier League for the start of the 2013-14 season.
And a trial - possibly in Britain - using video assistant referees for "game-changing decisions" will begin no later than the 2017-18 campaign, the International Football Association Board (IFAB) has said.
Middlesbrough Mayor Ray Mallon has referred himself to his council's standards committee in response to accusations a land sale was "dodgy".
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Thirteen climate change activists have pleaded not guilty to charges of trespassing on Heathrow's north runway.
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Southampton manager Claude Puel says he would like to see video technology used for offside after Manolo Gabbiadini's disallowed goal in the 3-2 EFL Cup final defeat by Manchester United.