Trump and Johnson might publicly announce the timeline when they meet in New York this week, the tabloid reported.
"The political will is there now on both sides to do the deal by July," a senior UK government source told The Sun. "It's a great win for us, and Trump is also really keen to shout about it in the States."
"There is also a recognition on both sides of the Atlantic that it must be done by then because the US election cycle starts soon afterwards," the source added.
Earlier, the British PM and the US President had planned on setting out a full timetable for the deal’s negotiation on Tuesday, but that has now been delayed, according to The Sun, at least until Trump’s next visit to London in December for a NATO summit.
Johnson has been eyeing a trade deal with the US since his race for the Conservative Party leadership back in July, naming it a key priority to overcome a no-deal Brexit. The PM is also set to meet with Germany’s Angela Merkel, France’s Emmanuel Macron, Irish Taoiseach Leo Varadkar and EU Council President Donald Tusk during his two-day visit to America for key Brexit talks.