(CNN) -- Despite some high-profile bombings in recent days, Iraq's security forces are ready to take over for U.S. forces this week to stabilize the nation's major cities, the U.S. commander in Iraq told CNN on Sunday.



Except for soldiers in advisory roles, all U.S. combat troops will leave Iraqi cities and towns by June 30.



Army Gen. Ray Odierno said he's seen a "constant improvement" in both the security situation and governance in Iraq to prepare for the June 30 deadline for U.S. troops to withdraw from major cities.



"They've been working for this for a long time," Odierno said on CNN's "State of the Union."



In a separate interview on "Fox News Sunday," Odierno said all U.S. troops already were out of Iraq's major cities before Tuesday's deadline.



"We have already moved out of the cities," Odierno said. "We've been slowly doing it over the last eight months. And the final units have moved out of the cities over the last several weeks."  Watch CNN's Michael Ware on the U.S. withdrawal »



The shift is part of the security agreement that former President George W. Bush's administration signed with Iraq.



In the CNN interview, Odierno blamed the recent violence in Iraq on "extremist elements using the timeframe and date to gain attention to themselves and divert attention from the success of Iraqi security forces."



The 131,000 U.S. troops in Iraq still will "maintain full coordination with Iraqi forces inside the cities" and continue to have intelligence capacity, Odierno said. With approval from the Iraqis, they also will carry out operations in major cities as necessary, he said.



Odierno said his goal is to help provide security that allows Iraq to hold planned national elections leading to the eventual removal of all U.S. troops by the end of 2011.



He said his biggest worry is a breakdown in stability such as a "consistent increase in violence" or a situation that Iraqi forces can't handle.



"I don't see that" happening, Odierno said. "I think we're on the right path."



Odierno also said Iran continues to "interfere" in Iraq, including training insurgents and paying surrogates. But he said his mission is limited to providing security within Iraq, no matter the provocation from Iran or elsewhere.



"I'm not authorized to do anything outside the borders of Iraq," he said.



Iran's government has repeatedly denied fomenting violence inside Iraq.
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