Several political supporters of the group have also been taken into custody.
More arrests are expected in course of an ongoing police operation that began with a swoop on a number of private residences in Srinagar, the summer capital of Kashmir state, and elsewhere at the weekend.
The detained people have been accused by the Kashmir police of plotting to mount daring "terrorist attacks" in several Indian cities as well as the Kashmir Valley. Among those held is a woman.
"They were planning some high-profile actions in the city of Srinagar as well as in Mumbai, Pune and New Delhi such as assassinating important political leaders and the police brass," Director General of Police (DGP), Gopal Sharma, said on Tuesday.
In Srinagar, the suspects were preparing to ram an explosives-laden vehicle into a VIP's motorcade, he said. Other police officers said Kashmir chief minister, Mufti Muhammad Sayeed, was a probable target of the planned attack.
Sharma said, "With the capture of 20 people, we have in fact busted the entire Lashkar-i-Tayyiba network operating from Srinagar for the last two years."
He said two of the 20 suspects were killed on Tuesday morning when they tried to slip away, taking advantage of a gun battle between the Kashmir police and armed fighters hiding on the outskirts of Srinagar.
The skirmish took place when a combined team of police, army and border guards was searching the area for suspected rebels, Sharma said, adding that five securitymen were injured in the encounter.
Sharma claimed the two slain suspects were Pakistani nationals - Shahid Ahmad and Zahid Hafiz - and said they were, respectively, divisional and district commanders of the Lashkar.
Sharma said the Lashkar cell was working with three independent modules: the Hassan, Usman and Rahman groups.
Policemen are a frequent target
of the armed separatist groups
According to Sharma, the Hassan group was responsible for detonating landmines and other improvised explosive devices; the Usman group specialised in carrying out surprise fidayeen attacks; while the Rahman group carried out political and other targeted killings.
The Lashkar cell had "strong links" with other groups fighting Indian rule in disputed Kashmir, such as Hizb al-Mujahidin and Al-Omar Mujahidin, according to Sharma.
To this end, they had launched a joint front called "Save Kashmir Movement" in whose name they would have claimed responsibility for the planned killings, he said.
"They were working on directions from across the border," Sharma said, but did not specify whether he meant Pakistani security officials or armed group leaders based in Pakistani-administered Kashmir.
The Kashmir Police DGP said one of the suspects, Shahid, had flown into New Delhi for the purpose and, in February, went on a "reconnaissance mission" to Mumbai and Pune - two important cities where he succeeded in establishing contacts.
"They were planning a fidayeen attack on the Mumbai Stock Exchange."
But one of the plotters, Babar - allegedly sent by Shahid to carry out attacks in Mumbai and Ahmedabad, the Gujarat city that was the scene of anti-Muslim violence two years ago - was killed by the local police, according to Sharma.
With regard to Kashmir, the alleged Lashkar fighters have been charged with the killings of a senior police officer, a ruling party leader, and an opposition party activist in addition to mounting half a dozen fidayeen attacks in Srinagar.
"With the arrest of more than 20 people, we have in fact busted the entire Lashkar-i-Tayyiba network operating from Srinagar for the last
Director General, Kashmir Police
One of the victims of those attacks was Javid Husain Shah, a prominent former Kashmiri separatist who later helped Indian security forces score several important victories in their counter-insurgency campaign.
Other victims included an official of an Indian government-run telephone company and several security men.
In a separate development, the Indian defence minister, Pranab Mukherjee, has arrived in the garrison town of Udhampur as part of a three-day visit to the militancy-hit region.
Thousands have perished
in the 15-year-old conflict