Obama's four picks will bring the Democratic president's nominees for the court to five of its 12 full-time judges.
One open spot is in Utah, because Judge Michael Murphy, of Salt Lake City, gave up full-time service last month.
Obama also will be filling openings on the six-state court from Wyoming, Kansas and Oklahoma. The Denver-based court also serves Colorado and New Mexico.
Of the 10 full-time spots now occupied, one was filled by President George H. W. Bush, three by President Bill Clinton, five by President George W. Bush and one by Obama.
After Obama picks nominees for the vacant seats, the Democratic-controlled Senate will consider confirmation.
Two spots one each from Oklahoma and Kansas are unfilled because Senate Republicans blocked Obama's nominees.
Obama's only nominee to the court who has received confirmation is Judge Scott Matheson Jr., of Salt Lake City, who the Senate confirmed in December 2010.
Carl Tobias, a University of Richmond law professor who studies and comments on the federal judicial appointment process, thinks Obama will avoid potential obstacles from Republican senators this year.
For the president to have his nominees for the Utah, Wyoming and Kansas spots confirmed, White House consultation with Republican senators from those states prior to making nominations will be critical, Tobias said.
"I think that Obama can fill these vacancies," just as he has filled large numbers of vacancies in other federal circuit courts, the professor said.
"Don't forget that Sen. Hatch favored Judge Matheson and shepherded him very effectively through the confirmation process and would do the same for a similarly qualified mainstream nominee," Tobias said.
Four new 10th Circuit judges would be a third of the active judges.
The court decides many noteworthy cases from Utah. Among recent ones are:
• A decision that a nuclear waste group comprised of several Western states, including Utah, has the authority to regulate storage of low-level radioactive waste and could prohibit EnergySolutions from importing foreign waste to its Clive, Utah facility.
• A decision that Utah doctors and child-protection workers did not violate the constitutional rights of Daren and Barbara Jensen when they pushed to have the Sandy couple's 12-year-old son, Parker, undergo chemotherapy.
• A decision that upheld the conviction of environmental activist Tim DeChristopher for making false statements and interfering with a gas and oil lease auction.