The change of heart comes in the wake of recommendations by Education Leeds which will be discussed at a Leeds City Council executive board meeting next Wednesday when a final decision will be made.
Education bosses are recommending the three schools remain open although plans to close four others are still set to go ahead.
The news was welcomed by Jan Tootill, headteacher at Beckett Park, one of the schools being offered a reprieve.
"We are absolutely delighted. Everyone worked so hard in the community and together we have made a difference.
"I'm pleased Education Leeds listened to what we had to say. When we were in limbo it was very difficult but now we can focus on making this a thriving community school."
Chris Edwards, chief executive of Education Leeds, said yesterday: "We have to address the issue of falling pupil numbers in primary schools across the city. There are now 6,000 fewer children in Leeds primaries than there were in 1997 and numbers will continue to fall by around 1,000 to 1,500 each year."
Mr Edwards said this meant the city's primary school budget, which is based on the number of schoolchildren, was losing about £2m a year in real terms.
"That is the equivalent of 60 fewer teachers, 120 fewer classroom assistants or five fewer primary schools."
He added: "In order to ensure our really wonderful primary provision remains effective and efficient we need to act and we need to make some very difficult decisions.
"We know that our primary schools are important to their local communities but there are simply not enough children to fill the places. We need to protect the vast majority of our provision by tackling these last remaining issues."
Coun Judith Blake, executive board member with responsibility for education, said: "These proposals are the result of lengthy consultation with schools, parents, community representatives and other organisations.
"Where areas are undergoing regeneration initiatives, this has been taken into account and we have been working to address such specific issues on an area by area basis."
If the proposals are ratified the original plans to close the three schools next August will be shelved and the governing bodies will instead look at ways of developing the schools.
But four other primary schools: Royal Park Primary School, Hyde Park; Langdale Primary School, Woodlesford; and Bentley Primary School and Potternewton Primary School, both in Meanwood, are still set to close next year.
When the proposed closures were first announced last November there was widespread opposition.