NEW DELHI/ISLAMABAD (Reuters) - India summoned Pakistan's high commissioner in New Delhi on Tuesday to protest against what it said was the continued infiltration of militants from across the border, an allegation denied by its arch rival.

India's Foreign Ministry said an arrested Pakistani national Bahadur Ali had confessed that he was "infiltrated into India" after being trained by the Lashkar-e-Taiba (LeT) militant group.

"The government of India strongly protests against the continued infiltration from Pakistan of trained terrorists ... this is contrary to assurances given by the Pakistani leaders at the highest level," the ministry said.

Pakistan's foreign affairs ministry rejected the Indian allegation, saying in a statement it was committed not to allow the use of its territory for any militant activity against anyone. Still, Pakistan said it would gather details to establish the veracity of India's claim.

Indian Interior Minister Rajnath Singh called last week for the isolation of countries deemed to be supporting "terrorism" in a speech at a regional conference in Islamabad.

Singh did not mention Pakistan by name but India accuses its neighbor of sheltering militants fighting against Indian rule in its part of divided Kashmir.

Earlier on Tuesday, the Indian foreign secretary met Pakistan High Commissioner Abdul Basit and issued a statement that India's Foreign Ministry spokesman described as a "strong demarche".

Ali had remained in touch with LeT's "operations room" before his arrest in July and received instructions to carry out an attack in India, the government said, adding it was prepared to grant the Pakistan High Commission consular access to Ali.

Relations between Pakistan and India have been strained further in recent weeks by a flare-up in protests against Indian rule in its part of the disputed border region of Kashmir, in which dozens have been killed by security forces.

The two countries have fought two of their three wars since independence in 1947 over Kashmir, which they both claim in full but rule in part.

India accuses Pakistan of training and arming militants in the portion it controls and sending them to the Indian side, a claim its neighbor denies.

(Reporting by Aditya Kalra in New Delhi and Amjad Ali in Islamabad; Editing by Tom Lasseter, Janet Lawrence and Adrian Croft)