A deadly uptick of Russian guided glide bombs slamming into Ukrainian cities — as many as 800 in a single week this summer — has injected new urgency into a long-running debate over whether Ukraine should be allowed to launch missiles supplied by the West at military targets deep in Russian territory.

Amid signs that President Biden is wavering, the issue will be on the table when he meets in Washington on Friday with Prime Minister Keir Starmer of Britain, after the two leaders dispatched their top diplomats to Kyiv on Wednesday to hear out the latest pleas from Ukraine’s president, Volodymyr Zelensky.

Ukraine has for months asked to use Western long-range weapons to attack more of the military sites that Russia uses to launch missiles and house the warplanes that drop the large, free-fall glide bombs that are wreaking havoc on Ukrainian forces and cities.

This past spring, Mr. Biden put specific limits — around 60 miles — on how far Ukraine can fire American-made surface-to-surface Army Tactical Missile Systems into Russia, leery of spurring a harsh retaliation from President Vladimir V. Putin.

U.S. officials said Britain and France, which gave Ukraine long-range, air-launched “bunker busters” in 2023, appear to be waiting for Mr. Biden’s endorsement before allowing those European missiles to hit targets far into Russian territory. The officials, like other U.S. officials interviewed for this article, requested anonymity because they were not authorized to speak publicly about the issue.

Mr. Putin’s spokesman, Dmitri S. Peskov, warned on Wednesday that Russia was preparing “appropriate countermeasures” should the West extend Ukraine’s authority, RIA Novosti, a Russian state-run news agency, reported.