Both sides have tried to "de-dramatise" the issue and are sounding more upbeat that a deal is in reach.
But Mr Gardiner said: "Whoever writes the question manipulates the answer that they want to get, and it will be this Government that is writing the question".
Johnson, May's former foreign minister, scolded the prime minister for her handling of negotiations on the future of the border between the British province of Northern Ireland and the Irish Republic, an European Union member state.
The scientific community is calling on her government to retain a close association with European science, in the form of a "closest possible associate membership" within the overall deal with the EU.
May said that if she got a deal, she thought parliament would ultimately approve it.
Under the plan, goods could be tracked using barcodes on shipping containers for "trusted trader" firms routinely importing and exporting between the two territories.
Meanwhile, Boris Johnson, the former foreign secretary, has once again attacked Mrs May's Chequers plan describing it as a "spectacular political auto crash".
No new border plan will materialise before the Conservative Party conference on September 30-Oct 3, the diplomat said.
Herald Sun doubles down on Serena Williams cartoon deemed racist
Another Twitter user, Paul Pellen, added: " Outrage for the sake of outrage!" However, he would go on to recant his observation. She then demanded an apology from the umpire, Carlos Ramos . "You are a liar ", Williams said , while pointing her finger.
Ahead of the meeting, Mrs May welcomed Malta's Prime Minister to Downing Street, talking Joseph Muscat through her Brexit blueprint in a bid to win support for it.
"But I can say at this point, without disclosing any numbers, is that clouds on the horizon have not become lighter but darker", she said.
The British Prime Minister's official spokesman said: "Boris Johnson was a member of the cabinet which agreed to the December joint report, including the backstop".
International Monetary Fund chief Christine Lagarde poured cold water on that notion Monday, saying "all the likely Brexit scenarios will have costs for the United Kingdom economy". "With time rapidly running out, we are left with two possibilities ― a bad deal, which could end up being so vague that we leave the European Union blind to what our future relationship will be, or a 'no-deal' Brexit".
The backing of Khan, a member of the main opposition Labour party, for a second referendum will put more pressure on Labour leader Jeremy Corbyn to change his opposition to the idea when the party meets for its annual conference in a week's time.
"There is a lot of talk about various issues around Brexit and we are anxious that in all these talks and negotiations, somehow science should not take a back seat", said Prof.
The government is set to publish its detailed immigration proposals in the coming months, which is widely expected to be a general toughening of stance on controlling the numbers that come into the country in line with the Conservative party's target to reduce migration.
With Mrs May lacking a parliamentary majority and Brexiteers in open rebellion, Labour could have huge a say in what happens next.