The Cosmonautics Museum in Moscow celebrating Soviet achievements in space. Sergey Ponomarev for The New York Times

MOSCOW — Russia’s Museum of Cosmonautics displays with rightful pride artifacts from its early years of storied achievements in space exploration: the first satellite, the first dog in space, the first man and, soon thereafter, the first toolbox.

Labeled in blocky Cyrillic writing a “panel with instruments for technical service and repair,” the toolbox held an array of handy items like pliers, two wooden-handled files and a hacksaw, spare blade included.

Russian space officials are touting this history of grit and ingenuity in orbit as they hope to persuade Washington to continue joint piloted exploration in the next decade rather than split into separate paths. They face significant hurdles.

The American incentives for engaging with Russia in space in the 1990s — political goals like the employment of idle rocket scientists to prevent missile proliferation — have mostly disappeared with the resumption of tensions.

The Trump administration has already proposed that by 2025 the United States should stop supporting the International Space Station that is the principal joint project today. A final decision is up to Congress. The American role might be shifted to a commercial footing thereafter.

In its place, NASA plans to place a habitable station called Gateway in orbit around the moon and send probes to the surface, while testing technologies for possible trips to Mars.

Further complicating matters are the plans by some entrepreneurs to launch private space stations for space tourism.

The talks between Russia and the United States promise to be difficult, and they have not been helped by a mysterious incident on the International Space Station. In August, a hole was discovered in the wall of a Russian Soyuz capsule docked to the station; air was leaking out.

Dmitri O. Rogozin, right, the director of Russia’s space agency, with President Vladimir V. Putin. “Politics ends where the opinion of a serious scientist begins,” Mr. Rogozin has said.

Dmitri O. Rogozin, the director of Russia’s space agency, said in an interview that the hole had been drilled in a deliberate act of sabotage, but it remains unclear whether this happened before launch or in orbit.

“It was intentional damage to the ship, we are convinced of this now,” Mr. Rogozin said of the hole, about the size of a pencil eraser. Cosmonauts on board patched it with specialized tape. “This was intentional action; manual, intentional action.”

Russian news media outlets have speculated wildly about NASA astronauts sabotaging the Russian capsule. The United States commander of the International Space Station at the time denied the accusations.

The capsule was launched in June but started leaking air only weeks later, suggesting that if the hole had been drilled before launch it must have been plugged with a sealant that broke down over time and was sucked into the vacuum of space.

On Tuesday evening in Moscow, two Russian cosmonauts, Sergey Prokopyev and Oleg Kononenko, exited the station for a spacewalk to investigate the exterior of the capsule. They planned to remove a panel from the craft and return it to Earth for an examination for signs of leaked sealant.

The investigation has to be conducted in space, as the portion of the Soyuz ship with the hole is designed to separate and burn during re-entry, meaning that…