AFTER BEING CLRED FOR TKOF; A B737 DEPARTING PHX OBSERVED AN ACFT XING DOWNFIELD.

Narrative

FLT WAS CLRED INTO POS FOR TKOF ONTO RWY 25R. THE TIME OF DAY WAS ABOUT MID AFTERNOON CENTRAL AND WX CONDITIONS WERE VFR WITH EXCELLENT VISIBILITY. PRIOR TO COMING TO A COMPLETE STOP ON THE RWY; WE WERE CLRED FOR TKOF. IT WAS THE FO'S TKOF; AS HE ASSUMED CTL OF THE ACFT AND BEGAN TO PUSH THE PWR UP; WE BOTH NOTICED ANOTHER CARRIER ACFT APCHING RWY 25R FROM OUR R. THEY APPEARED TO BE ON EITHER TXWY S OR T; A FEW THOUSAND FT DOWN THE RWY FROM OUR POS. IT DID NOT APPEAR THAT THEY WERE GOING TO HOLD SHORT OF OUR DEP RWY. OUR ACFT WAS ROLLING FORWARD AT A VERY SLOW SPD; MUCH LESS THAN NORMAL TAXI SPD. FO HAD BARELY MOVED THE THRUST LEVERS FORWARD WHEN WE NOTICED THE OTHER ACFT. WE COMMUNICATED OUR OBSERVATION OF THE ENCROACHING ACFT TO EACH OTHER AND DECIDED TO CONTINUE WITH OUR ACFT SLOWLY MOVING FORWARD. AS THE OTHER CARRIER ACFT BEGAN TO CROSS THE RWY THEY RADIOED TO PHX TWR THAT THEY WERE XING RWY 25R; WHICH THEY DID IN A VERY EXPEDITIOUS MANNER. AS THEY CLRED THE RWY TO THE S; I RADIOED TO PHX TWR SOMETHING TO THE EFFECT; 'I GUESS WE ARE STILL CLRED FOR TKOF?' THERE WAS NO REPLY FROM THE TWR. THE RWY WAS NOW CLR; WE HAD BARELY ROLLED FORWARD FROM OUR ORIGINAL STARTING POS; SO WE CONTINUED WITH OUR DEP. THE REMAINDER OF THE FLT PROGRESSED UNEVENTFULLY. UPON ARRIVING IN OAKLAND I CALLED PHX TWR AND SPOKE WITH THE SUPVR ABOUT THE EVENT. HE INDICATED THAT THE PROB WAS FROM WITHIN THE TWR. I ALSO CALLED OUR 'AT' SPECIALIST; WHO ALSO CALLED PHX TWR; AND CONFIRMED THAT WE HAD DONE NOTHING WRONG AND THAT THE PROB OCCURRED WITHIN THE TWR.