The government is under pressure from two influential House of Commons select committees to drop the boycott of contacts with Hamas which it shares with members of the international Middle East Quartet.
The chairman of one of the committees, Labour MP Mike Gapes, last week held private talks on the issue with Tony Blair, Middle East envoy of the Quartet - the United States, the European Union, the United Nations and Russia.
The Quartet has pledged to refuse dialogue with Hamas until it recognises Israel's right to exist, renounces violence and agrees to abide by all agreements between Israel and the Palestinian Authority.
Mr Gapes, chairman of the foreign affairs committee and a noted supporter of Labour Friends of Israel, said: "The Quartet, including Britain, should try to engage with moderate elements within Hamas, especially in view of the current Gaza ceasefire."
A similar message came from the International Development Committee last week which, in a report on conditions in the Occupied Territories, called for an exploratory sounding-out of Hamas.
But one committee member, Tory Stephen Crabb, MP for Pembrokeshire, and chairman of the Conservative Party Human Rights Commission, said the recommendation was "over the top".
The Foreign Affairs committee said that a dialogue could allow the international community to exert effective pressure on Hamas to stop abusing human rights. It added: "Some of Israel's actions against the Palestinians have been disproportionate and we conclude that Israeli policies towards the population of the Gaza Strip have been collective punishment."
Malcolm Bruce MP (Lib Dem), chairman of the IDC, said his committee understood the Quartet's conditions. "But there cannot be a credible peace deal without involving Hamas."
There was a cool response from Israeli diplomats. An Israeli embassy spokesperson said: "Israel has proved in the past that it is more than willing to talk with enemies who want peace and are willing to accept Israel's right to exist.
"However, this is not the case with Hamas, [which] is determined to stay irrelevant by not accepting even one of the principles set by the international community. Hamas' actions speak louder than words, and Hamas has proved it is no partner for dialogue."
It was a pity, she added, that the parliamentary committees failed to realise that "Hamas is part of the problem and not part of the solution."