Other countries are "queuing up" to sign trade deals with the UK once it leaves the EU, Boris Johnson has said.
Mrs May set out her Brexit strategy - which includes pulling out of the EU single market - in a speech on Tuesday.
She also warned the EU not to try to "punish" the UK, saying she would walk away unless the right deal was offered.
In his first reaction to the speech, European Commission President Jean-Claude Juncker promised to work to ensure Brexit talks are carried out "according to the rules and they yield good results".
He added: "I welcome the clarifications given by Mrs May, but I said to her last night that a speech will not launch the negotiations."
With just over two months to go before the UK government is due to get talks under way, Mr Johnson wrote in the Daily Telegraph that the UK would "no longer have our trade policy run by the EU commission".
He added: "That means - crucially - that we will be able to do new free trade deals with countries around the world. They are already queuing up.
"Under EU rules, we are not formally allowed to negotiate these new treaties until we leave. But there is nothing to say that ideas cannot be pencilled in."
EU leaders have begun to deliver their verdicts on Mrs May's speech.
Before Mr Juncker spoke, Maltese Prime Minister Joseph Muscat, whose country currently has presidency of the EU, warned: "We want a fair deal for the United Kingdom, but that deal necessarily needs to be inferior to membership.
"Thinking it can be otherwise would indicate a detachment from reality."
The Czech Republic's Secretary of State for EU Affairs, Tomas Prouza, told BBC Radio 4's Today programme it was important that a Brexit deal "makes sense for both sides".