The US and Israel launched air strikes on Iran over the weekend and the conflict has led to airspace closures and widespread flight cancellations across parts of the Gulf, disrupting a key transit hub.
Former world number one Medvedev, who won the Dubai Championships title via a walkover on Saturday, had been scheduled to play in the Eisenhower Cup, a one-night Tie Break Tens doubles event alongside fellow Russian Mirra Andreeva.
American Amanda Anisimova had been paired with Rublev for the event on the eve of the main draw of the Indian Wells tournament in the Southern California desert.
Medvedev, a two-time finalist at the tournament who is seeded 11th, and 17th seed Rublev are both scheduled to play their first singles matches on Friday.
Challenger event halted
Fallout from the conflict also led to the abrupt halt and cancellation of an ATP Challenger event in the United Arab Emirates on Tuesday due to a security alert, causing players and tournament personnel to flee from the courts.
The Professional Tennis Players Association (PTPA) said the ATP Tour was offering players the option of a charter flight at 5,000 euros ($5,800) per person, paid out of pocket, to leave the country.
PTPA offered to pay half of the cost for the players and called on the ATP Tour to cover the rest.
"We are taking this step to ensure that no player is forced to choose between their safety and their financial stability," the PTPA said in a statement.
The ATP Tour did not immediately respond to a request for comment but Belarusian player Ilya Ivashka, who was among those stranded in the city, posted on social media that the men's governing body had stepped up to help them.
"Just got confirmation that all the people from the tournament in Fujairah will be evacuated with no cost," Ivashka posted on Instagram.
"Everything completely covered. Class act @atptour."
Gauff's coach stranded
American Coco Gauff said her coach Gavin MacMillan was among those stranded in the Middle East.
"He's currently stuck, and I don't know if he's going to make it," she told a press conference. "I just want him to be safe."
Briton Jack Draper, the Indian Wells defending champion, described the situation as very concerning.
"I was in Dubai last week. I managed to get away almost probably one of the last flights out," he said.
"I just hope the players and all the staff within the ATP are able to make it here. The main thing is that they're safe."