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A post-Brexit British government should focus on its new relationship with the European Union before cementing ties with the U.S., a former White House official told CNBC on Tuesday.

"There's definitely an interest," from the U.S. in cementing the relationship with the U.K., Tony Fratto, founding partner at Hamilton Place Strategies and former deputy press secretary to President George W. Bush, told CNBC in London.

"The UK has other business ahead of that right now which is figuring out its relationship with Europe before it figures out its relationship with the United States," Fratto said, adding that the EU-UK trade represents 40 percent of the U.K. economy.

"They need to focus on that first," he reiterated.

After the Brexit vote and with key elections coming up, the European Union is facing new challenges over its unity and political stability.

Fratto told CNBC he hoped that the EU would not fall apart.

"From the U.S. perspective, a unified Europe actually makes a lot of sense. We preferred to see the UK as part of that," he said.

Meanwhile, the U.S. has business of its own to deal with. President-elect Donald Trump continues choosing the names for the new cabinet.

"I think we will see a lot of issues moving more than we have seen in the past few years." Fratto told CNBC, mentioning tax reform as one of the most pressing issues.

"They're anxious to get to it," he said.