Heads' leaders are in talks with Government officials about allowing schools to opt out of controversial synthetic phonics reading tests.
The NAHT union is arguing that primary schools should be excused from the Year 1 tests if they can show that their existing reading checks cover synthetic phonics.
NAHT general secretary Russell Hobby said Department for Education officials had given their proposals a "reasonable reception", with more meetings to come.
"If a school has better ways of delivering the objectives then they should be enabled to do so, instead of using the tests," he said.
The NAHT believes schools can develop schemes of their own that are more sophisticated than the proposed 10-minute screenings.
"If you read with pupils on a regular basis you can pull out results that look at comprehension as well as phonics," Mr Hobby said. "It could give you a more sophisticated summary of a child's abilities and be much more informative for parents."
The Government's scheme will be piloted in 300 schools this summer, with plans to introduce the screenings nationally in 2012.
Pupils will be tested one-to-one with a teacher on 40 items. Non-words like "zort" and "koob" will be included alongside real words to ensure children are reading, rather than just memorising.
Schools minister Nick Gibb said: "We are developing how the check will work and are taking on board the views of all interested parties."