With Ukraine’s second-largest city bracing for a new Russian offensive, a growing number of NATO allies are backing Kyiv’s pleas to allow its forces to conduct strikes in Russian territory with Western weapons.

President Biden has decided to let Ukraine use American weapons against military targets in Russia to blunt the Kharkiv offensive, days after Canada decided to allow the use of arms it has supplied. More than a dozen have given similar permission to Ukraine.

The United States, the most important supplier of weaponry to Ukraine, had been reluctant to take the step, worried about provoking Russia into an escalation that could drag in NATO and set off a wider war. Without sign-off from Washington, the American-made Army Tactical Missile Systems, or ATACMS, can strike Russian targets only inside Ukraine.

Yet many Western leaders and military analysts say that with Russia massing thousands of troops on its side of the border — less than 20 miles from the northeastern city of Kharkiv — Ukraine badly needs the authority to strike inside Russia with Western weapons. The permission from President Biden is intended solely for Ukraine to attack military sites in Russia being used for the Kharkiv offensive, U.S. officials said.

“Russian commanders are well aware of Ukraine’s inability to strike back,” Peter Dickinson, a Ukraine analyst at the Atlantic Council in Washington, wrote in an analysis published before Mr. Biden’s policy change.