At a training session of NUT representatives in Nottingham last week, members were angry that, despite being pressed, Mr Sinnott did not confirm that the boycott will definitely continue.
He said the union was due to meet Gillian Shephard, the Education Secretary, before Christmas and the union's response would depend on any concessions made by her.
John Illingworth, treasurer of Nottinghamshire NUT, said: "His reluctance to confirm total support for the boycott made the meeting very angry and made us suspect that the national executive is wobbling. The local view is that most teachers fully support the boycott. This year is particularly important because of key stage 2, which many in the union believe is a backdoor 11-plus.
"It is not just NUT members - teachers from other unions are happy to avoid the SATs by clinging to the coat-tails of our boycott; they too will be bitterly disappointed if the action is stopped."
A recent survey carried out by the union found that those who replied were strongly in favour of continuing with the boycott and were vehemently opposed to league tables.
Mr Sinnott acknowledged the feelings demonstrated in the survey, but suggested that a certain amount of flexibility was possible in discussions with the Education Secretary.
Mr Illingworth said he thought that Doug McAvoy, the union's general secretary, was holding the boycott together, but believed that members of the Broad Left faction, who supported Steve Sinnott, may be less likely to be persuaded by Mr McAvoy to support the continuation of the action now their own candidate is deputy general secretary.