Russia has massed troops on Ukraine's border, and Putin has said that the United States and NATO have not addressed Moscow's main security demands, though he was ready to keep talking.
The United States says Russia now has the capability to act against Ukraine, and Johnson has warned of a catastrophe from which no-one would win if Russia were to invade.
Britain has said it will sanction Russia if it makes any incursion into Ukraine, and has urged European allies to prepare similar steps. His government is expected to announce a toughening of its sanctions regime on Monday.
"The Prime Minister is determined to accelerate diplomatic efforts and ramp up deterrence to avoid bloodshed in Europe," a spokesperson from Johnson's Downing Street office said in a statement.
"He will reiterate the need for Russia to step back and engage diplomatically when he speaks to President Putin this week."
Johnson's office did not give details of when or where the trip would happen.
Britain has supplied defensive weapons and training personnel to Ukraine, though ministers have said that the deployment of combat troops is unlikely.
Johnson is considering options including fresh deployments and the bolstering of NATO defences, and on Monday Britain will join U.N. Security Council discussions to try and resolve the standoff through diplomatic means.
(Reporting by Alistair Smout; Editing by Daniel Wallis)