Clinton has 39% support to Trump's 36%, with Johnson pulling 12% backing, according to the survey of 1,004 registered voters, which was conducted June 5-8.
With Johnson dropped from consideration, Clinton's lead remains three percentage points -- 42% to Trump's 39% -- the poll found.
Both candidates are viewed negatively by large chunks of the electorate.
Clinton's favorability rating is 42%, compared to an unfavorability rating of 56%. Trump is viewed favorably by 40% of the voters surveyed, while 58% view him unfavorably.
The Fox News poll, conducted by the Democratic firm Anderson Robbins Research and the Republican firm Shaw & Company Research, found much stronger ratings for Vermont Sen. Bernie Sanders, the primary rival who Clinton recently vanquished to claim the party's title as presumptive nominee. Sanders is viewed favorably by 54% and unfavorably by 41%, and tops Trump head-to-head, 49% to 38%.
The poll's margin of error is plus or minus four percentage points.
It found Clinton with some big advantages. Asked if she has the knowledge to serve effectively as president, 71% said yes and 27% no, compared to 38% and 59%, respectively, for Trump.
And asked about their temperaments, 60% said Clinton has the temperament to serve effectively as president, compared to 38% saying no, while Trump's numbers were 33% saying yes and 62% saying no.
Trump, meanwhile, got much better marks when asked who would shake up Washington. Seventy-six percent said he would, to just 21% said he wouldn't. For Clinton, 37% said she would shake Washington up while 59% said she wouldn't.
The survey also found Clinton's use of a private email server during her tenure as secretary of state to be a major liability. Asked if they believe she is lying about how her emails were handled, 60% said yes and 27% said no; 57% said she put national security at risk by mishandling her emails.