The Sunday Times newspaper said the "Boris bounce" had pushed support for Johnson's Conservatives to 31%, up six percentage points from the previous poll, while Labor was on 21%, up two points.
The newspaper said the poll by YouGov could fuel speculation that Johnson, who faces a Brexit deadlock in Britain's parliament, will call an early election.
The poll gave the Conservatives their biggest lead over the opposition in five months and it was the first time their support has risen above 30% since April, the Sunday Times said.
The Liberal Democrats were down three points on 20% and the new Brexit Party, led by veteran Eurosceptic Nigel Farage, was down four points on 13%, half its peak level in May.
YouGov questioned 1,697 adults on July 25 and 26.
A second poll showed a smaller increase in support for the Conservatives.
The poll by ComRes showed Johnson's party with 28%, up three points and only one percentage point ahead of Labor.
The ComRes poll, conducted for the Sunday Express newspaper, showed the Conservatives would fall further short of a majority in parliament than they are now.
While preferred as a leader to Labor's Jeremy Corbyn, most voters think Johnson will be a bad prime minister, ComRes said, Reuters reported.
"As Boris Johnson begins his premiership we have seen an anticipated bump in Conservative support, generally at the expense of the Brexit Party," Chris Hopkins, head of politics at ComRes, said.
"However, while the public agree that he should be given the necessary time to deliver Brexit, a majority are skeptical as to how good he may be as prime minister."
If the poll figures were replicated at a national election, the Conservatives would be the largest party but would fall 48 seats short of a majority, ComRes said.
The poll was conducted for the Sunday Express newspaper and interviewed 2,029 British adults on July 24 and 25.