Heads paid pound;100 to disclose results
The National Assessment Agency (NAA) said it was collecting the results from Years 3, 4 and 5 to check how pupils are doing in the years they do not sit national tests.
But the National Association of Head Teachers is concerned that the information may be used to compile league tables - a claim the agency has denied.
Mick Brookes, the association's general secretary, said: "Our strong advice to schools is not to do it because we have no trust in the use of this data ... We are concerned that this could lead into constructing league tables for every year group. We can't allow that to happen.
"There is also the issue of the imposition of bureaucracy. Is it something that's going to benefit the school? If not, then don't do it."
The agency said the Department for Children, Schools and Families required it to track national trends.
ETS, the company that is overseeing national tests this year, has been collecting the data.
An agency spokesman said: "A representative sample of schools has been identified in the past and they were approached to ask whether they would provide optional test data. The process is effectively unchanged from previous years."
Sue Fielden, head of Torrisholme Primary and executive head of Skerton Primary in Lancashire, said she had been contacted by ETS but not offered any money.
"I wouldn't give them my results because they are marked within school and there is no moderation process," she said. "We use them in school for assessing the next steps in learning for individual children. They are nothing to do with anyone outside the school."
Meanwhile, NAA staff have been drafted in to sort out problems with this year's Sats. One marker said the agency had phoned to check her progress, while others said it had investigated their complaints.