The 1.7-million-member National Education Association -- backbone of the last three Democratic presidential campaigns and the bane of many of President Reagan's education policies -- has begun a rapprochement with the administration.

NEA President Mary Hatwood Futrell, in her keynote speech yesterday to the union's 123rd annual convention, ceased her strong attacks on the administration's education and tax policies long enough to ask Reagan to "join the NEA."

"Mr. President, we have disagreed with you in the past," Futrell said before the 8,000 delegates at the Washington Convention Center. "And we undoubtedly will disagree in the future. But, as the Bible tells us, there is a time to rend and a time to heal."

She added, "Mr. President, join us, for the children's sake, and let's work together for the public schools America needs."

That olive branch follows four years of often-bitter quarreling between the traditionally liberal union and the administration, which has advocated merit pay for teachers, school prayer, tuition tax credits and vouchers for parents of private school students -- all policies the NEA strongly opposes.

Reagan and the NEA also have sparred over the union's controversial curriculum on nuclear war. Most recently, in a news conference Friday, Futrell promised to "pull out all the stops" to defeat elements of Reagan's tax plan that she believes will harm public education.

Futrell's efforts at rapprochement yesterday come as she tries to shift her union toward a more moderate stand on issues affecting the profession. In her speech yesterday, she urged delegates to modify the NEA's longstanding opposition to standardized testing and support a national examination for all new teachers.

"Let us tell America," Futrell said, "that just as no law graduate can practice law without passing the bar exam, no teaching graduate should be allowed to instruct America's children without first passing a valid exam that tests mastery of subject matter and professional skills."

The NEA has lately been under attack by New Right groups and by critics who see the union as an obstacle to educational reform. Futrell's efforts at this convention have been aimed partly at putting the NEA in the forefront of reform efforts, as she calls for tough standards for teacher training schools and continuing evaluation -- but not testing -- of current teachers.

In her address yesterday, Futrell also proposed a $1-a-member dues set-aside, for a total of $1.7 million, to help combat illiteracy and the escalating school dropout rate. Called "Operation Rescue," it would give $700,000 to local dropout prevention programs designed by teachers. The remaining $1 million would be used for future programs.