"The process of reviewing information about potential nominees continues," said White House spokesman Josh Earnest, amid speculation that the appointment may come soon.
Obama's candidate would tilt the balance of the nine-member court currently evenly split between eight liberal and eight conservative justices. But that has prompted Republicans to say the appointment should be made by the next president.
The Republican-controlled Senate must approve any nominee before he or she can take up the post. Earnest said the announcement would "not necessarily" be made before Obama leaves for landmark trip to Cuba just over a week from now.
The White House has busied itself galvanizing Democratic support and turning up the heat on individual Republican Senators to allow a hearings. The issue is sure to energize the base of both parties as the country heads to a presidential election in November.
Traveling to Texas on Friday, the White House zeroed in on home state Senator John Cornyn and his pledge to treat Obama's nominee like a "pinata."
"The president hasn't even decided on a nominee yet. It sounds like Republicans are prepared to follow through on their threat... (and) attempt to tear that person down," Earnest said. "It's clear that Republicans vow to treat the president's nominee like a pinata is not motivated by some sort of principle, but rather is motivated by party politics."