The Government is coming under increasing pressure to sever ties with the American company which has overseen the marking of this year's national tests.
Details are emerging of problems with the quality of this year's marking.
Schools are now getting back pupils' scripts following the administrative failings that delayed the publication of the results.
But there are now reports of schools being sent back unmarked papers and of some pupils who took the tests being recorded as absent.
Shadow children's secretary Michael Gove called on the Government to scrap the contract with ETS Europe before next year's exams.
He said: "ETS have forfeited the right to run future SATs tests. Every day brings new evidence of their serial incompetence.
"Ministers must act now to ensure next year's exams are run properly. That means guaranteeing that a proper team to supervise the process is in place as soon as possible."
At a select committee meeting, Schools Secretary Ed Balls declined to apologise for the fiasco, saying the situation was "unacceptable" and he was "upset" at what had happened.
One primary school teacher was so upset by this year's marking she released examples of pupils' marked English papers.
Janis Burdin, of Moss Side Primary School in Leyland, Lancashire, said: "Reading the papers we were immediately aware they had been marked in an extremely erratic fashion. Some children had more marks awarded than they should have and some far less."
