The comments are a major boost to the Prime Minister with continued speculation over a Brexiteer led coup to install Boris Johnson as Prime Minister.
Dr Fox told Ridge On Sunday on Sky News: “I think the Prime Minister is doing a great job in difficult circumstances domestically because of the lack of a majority in the Commons and in the difficult task of this negotiation with the European Union.
“I think that supporting the Prime Minister now is in our national interest. Certainly if she wants to continue on to the next election she will have my support and I think a British Prime Minister that delivers a successful Brexit will have the support of the British public.”
Dr Fox also suggested that Brussels has made “reassuring noises” in recent weeks in the Brexit negotiations.
Dr Fox previously put the chances of a deal at 60-40.
Asked about the assessment, he told Sky News’ Ridge On Sunday: “Over the last few weeks we have certainly heard more reassuring noises from the European Commission and some of our European partners.
“They have understood that Britain is serious that if we don’t get a good deal we could end up with no deal.”
Dr Fox said the Chequers plan was a “reasonable, constructive way forward” in the Brexit talks.
“I do detect that there is a movement from the European Union towards getting a deal,” he said.
“It’s by no means certain but I think that it is moving in the right direction, although no-deal remains a possibility for which we have to prepare.”
Dr Fox dismissed suggestions that extending Article 50, which takes Britain out of the EU, would help secure a good deal.
“Extending Article 50 until we have got an agreement is effectively allowing the European Union to dictate when Britain will leave the European Union itself,” he said.