Johnson spoke with the embattled Myanmar leader, whose reputation among the international community has plunged over her handling of the crisis, in Naypyidaw while on a four-day tour in Asia.
The meeting followed Johnson's visit to a refugee camp in Bangladesh's Cox's Bazar district, where nearly 700,000 Rohingya have sought sanctuary in squalid settlements since a Myanmar army crackdown in northern Rakhine last August.
But Myanmar has staunchly denied the charges and blocked UN investigators from the conflict zone, souring relations with a host of western allies.
Fresh reports of mass graves in Rakhine -- and the arrest of two Reuters journalists investigating an alleged massacre -- have heaped new pressure on Suu Kyi to condemn the army, who she is in a delicate power-sharing arrangement with.
But the Nobel laureate has refused to change tack and is accused by critics of bunkering down in a siege mentality.
Ahead of the talks the UK's foreign office said Johnson would press for an "end to the suffering in Rakhine and the safe and voluntary return of the refugees".
Johnson is scheduled to visit Rakhine later on Sunday.
Myanmar and Bangladesh have inked a deal to bring refugees back to northern Rakhine, but repatriation has yet to begin.
Many Rohingya do not feel safe returning to a country where they have faced violent persecution and decades of discrimination at the hands of a state that has denied them citizenship.
Johnson is scheduled to fly on to Bangkok Sunday for a two-day visit that will include meetings with junta chief Prayut Chan-O-Cha and the Thai chairman of an advisory board on the Rohingya crisis.
Veteran US diplomat Bill Richardson dramatically stepped down from the board last month, saying he could not in "good conscience" sit on a panel he feared would only "whitewash" the causes of the Rohingya crisis.
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