"U.S.-Russian relations are important. It's important for stability, it's important for our relations in Europe," Bush said at a news conference.
"Therefore, my advice is to establish a personal relationship with whoever is in charge of foreign policy in Russia," the U.S. president said.
Presidential polls are due in Russia on March 2. First Deputy Prime Minister Dmitry Medvedev, who was publicly backed by President Vladimir Putin as his successor in mid-December, is a clear front-runner.
Putin, who is unable to stand for a third term under the Constitution, is expected to become premier if Medvedev wins the polls, which is almost certain to happen.
Earlier media reports said both Democrat hopefuls Hillary Clinton and Barack Obama failed to pronounce the Kremlin hopeful's name correctly.
"I want to try to leave it so whoever my successor is will be able to have a relationship with whoever is running foreign policy in Russia. It's in the country's interest," Bush said.
"There's areas where we need to cooperate."
Bush who is nearing the end of his presidential term with elections due in November admitted that he had disagreed with Putin in the past.
"As you know, Putin's a straightforward, pretty tough character when it comes to his interests, well so am I," Bush said.
Meanwhile, the U.S. leader added that the countries' bilateral relations remained "cordial" enabling them to "deal with common threats and opportunities, and that's going to be important for the next president to maintain."
Speaking on the U.S. missile shield plans in the Czech Republic and Poland, which has become another irritant in Russia-U.S. relations, Bush said: "I believe it's in our interests to try to figure out a way for the Russians to understand the system is not aimed at them, but aimed at the real threats of the 21st century."
