Mr Bone told BBC Newsnight: “I was amazed that how the one thing that she was fired up about and the one thing that was definitely going to happen is Brexit.
“And her eyes sparkled at the thought of it.
“I thought wow this is just what we need and if there was anyone that had any doubt they should have been a that meeting.”
He added: “If she delivered Brexit, and I think she will, she will be a national hero.
“And then why not carry on? I have not heard a single person of all my backbench colleagues saying we have to get rid of the Prime Minister, we have to have a new leader, it has to be he or she.
“It just isn’t there. There is no plotting. By the way I think if was plotting going on I would probably know about it.”
His comments come after a Tory MP told the Mirror that Mrs May would be stepping down from her post after Brexit, setting the date as August 30, 2019.
Theresa May delivering Brexit will make her a national hero, Conservative MP Peter Bone declares
Her eyes sparkled at the thought of it… and if she delivered Brexit, and I think she will, she will be a national hero
The alleged deadline would see her lead for five months after Britain leaves the EU on 29 March 2019, the nation’s official departure date.
But Mrs May has hit back at the rumours, saying she is in it for the long term.
Speaking to BBC on Wednesday, the Prime Minister said: “Yes, there’s been an awful lot of speculation about my future which has no basis in it whatsoever.
“I’m in this for the long term.
“There’s a real job to be done in the United Kingdom.”
Theresa May says she will stay on as Prime Minister – declaring she is in it for the long term
According to reports party members throw their support behind Brexiteer figureheads – including Brexit minister David Davis – and eurosceptic rising star Jacob Rees-Mogg.
Last week bookmakers slashed the odds on Mr Rees-Mogg becoming the next Tory leader, as his supporters continue to rally behind the ‘Moggmentum’ campaign.
The MP is the 5/1 second favourite to replace Mrs May with bookies William Hill, after being 33/1 just one month ago.
The eccentric backbench MP for North East Somerset, who is pushing for a hard Brexit, is just behind Cabinet minister David Davis at 10/3.
Chancellor Philip Hammond is tipped at 7/1 to take over from Mrs May, while Foreign Secretary Boris Johnson is fourth favourite at 8/1.
Mr Johnson in June revealed he will not challenge the Theresa May for Prime Minister until after Brexit – when rumours began circulating of a possible leadership contest.
The Foreign Secretary told Channel 4: “Yes we’ve got to get on and deliver the priorities of the people.
“What the country overwhelming wants is calm and stability and they want Theresa May to get on with her job.
Boris Johnson earlier dismissed claims he would challenge Theresa May for Prime Minister
“Whether you like or not we were entrusted… to deliver.”
When asked whether Mrs May will remain Prime Minister, Mr Johnson said: “What I tell you is that Theresa May got more votes than anybody since Margaret Thatcher in 1983.
“She leads the largest party in Parliament by a margin of 56 seats. It is her duty to go on and what she can.
“I think it was an excellent Queen’s Speech.”