The districts were Ganderbal, Badgam, Bandipora and Baramulla, police confirmed.
The state government has already ordered opening of schools in these four districts.
Officials said curfew and restrictions would, however, continue in parts of Srinagar city.
There was some civilian activity in parts of Srinagaron Saturday as pedestrian and private transport movement was seen in the City’s uptown areas.
Union Home Minister Rajnath Singh currently on a two-day visit to the Valley met many delegations on Saturday at the Nehru Guest House in Srinagar.
Official sources said these delegations included members of the civil society, clerics, Sikh, Pandit community members and traders.
Important local trade and industry bodies including the Kashmir Chamber of Commerce and Industry (KCCI) and Kashmir Economic Alliance (KEA) decided not to meet the Union Home Minister despite an invitation from him.
Reports said Rajnath Singh told the delegations that the ongoing unrest was triggered by “forces from across the border”.
“While deciding to evolve a mechanism to engage the people of Kashmir in a dialogue process to work out solutions for the problems faced by them, the Union Home Minister does not favour an engagement with the separatists who take their directions from across the border,” official sources said.
Rajnath Singh had a detailed discussion with state Governor N.N.Vohra.
He is slated to meet meet Chief Minister Mehbooba Mufti later on Sunday.
Before leaving for Delhi, the Union Home Minister will chair a high level security meeting to discuss the prevailing law and order situation, officials said.
Political party delegations scheduled to meet Rajnath Singh on Sunday include one from the National Conference (NC).
However, the Congress party has decided not to call on the minister.
G.A.Mir, president of state Congress unit stated that there was no point meeting Rajnath Singh saying, “There has not been a single people friendly measure from the state government since the Chief Minister chaired an all party meeting which was attended by us (Congress).”
In the meantime, four patients suffering from serious eye injuries due to pellet gun shots were airlifted on Saturday for specialized treatment at the All India Institute of Medical Sciences (AIIMS), New Delhi.
Train services between north Kashmir’s Baramulla town and Bannihal town in the Jammu region, however remained suspended for the 16th day on Sunday.