KALAK, Northern Iraq (CNN) -- Thousands of U.S. airborne troops are expected to fly into northern Iraq from eastern Jordan over the next seven days, bypassing Turkish airspace, Kurdish sources tell CNN. 

Up to 200 U.S. Special Forces will set up a staging area in eastern Jordan near the border with Iraq, the sources said, where they will fly the airborne troops into the region. These could number thousands after a week. 

Sources say the operation is the result of American exasperation with Turkey's demands that it should be able to put its troops into northern Iraq at same time as U.S. troops would go into the region. 

The Turks contend they need troops in northern Iraq to "manage the humanitarian situation" -- partly by keeping Kurdish refugees from entering Turkey. 

There are conflicting reports whether Turkish troops have already crossed the border into Iraq. 

Turkey agreed Friday to open two air corridors over that nation for U.S. planes to use for air attacks on Iraq, said Turkish Defense Minister Vecdi Gonul. 

Within hours, a contingent of more than 1,000 Turkish troops began crossing the border into northern Iraq, CNN Turk Military Correspondent Kemal Yurteri reported. 

But Kurdish sources disputed that figure, and a senior U.S. State Department official told CNN that American officials were not told of any troop movements and considered the issue "still under discussion." 

U.S. Defense Secretary Donald Rumsfeld acknowledged the Turks "have had some forces in northern Iraq for some time, not associated with what's going on right now. But in terms of any large numbers, they are not." 

The U.S. does not want Turkish troops moving into Kurdish-controlled areas of northern Iraq for fear of clashes between Turkish and Kurdish troops.