The unprecedented meeting would follow Monday's talks between Mr Trimble and other party leaders in Northern Ireland about the workings of the proposed new assembly. Reliable sources said the one-to-one talks could take place as early as next week.
Mr Trimble is believed to be on the point of consulting senior colleagues about taking the step. Although he might avoid the public handshake sought by the Sinn Fein president, the planned meeting is highly significant.
As First Minister of the new Northern Ireland Assembly, Mr Trimble is expected to hold a series of separate sessions with the individual party leaders in the province. He will now include Mr Adams in that series. Tony Blair has privately encouraged such a meeting, which he believes would give further momentum to the peace process.
On a day of important developments in the peace process, Sinn Fein finally agreed to appoint a contact with the international body overseeing the decommissioning of arms - the party's chief negotiator, Martin McGuinness.
The Sinn Fein announcement on decommissioning was welcomed by the British and Irish governments. Mr Blair described it as "a practical and important step forward" and said the decommissioning of terrorist weapons within two years was a vital part of a lasting settlement. "I hope the process begins as soon as possible," he said. Mr Blair told MPs there would be many more difficulties along the way, but added: "I have no doubt things are on the move and we are moving in the right direction."
British officials are surprised by the progress that has been made in the peace process so soon after last month's Omagh atrocity, which claimed 28 lives. Sources said it was hoped Mr McGuinness's appointment would reassure Unionist opinion, because it was "a firm deed rather than playing word games".
Sinn Fein refused to be drawn on whether the appointment should be seen as a sign of an IRA willingness to start disarming. But it said it hoped it would push the peace process forward with increased speed.
The Irish government described the appointment of Mr McGuinness as a further important step in the implementation of the Good Friday agreement.
Mr Trimble also welcomed the appointment but said he would urge republicans in next week's talks to "take the next necessary step of actual decommissioning". He warned: "Republicans can play no part in Northern Ireland's government if Unionists do not trust them. That trust can only start to be built by the commencement of decommissioning and an ending of all forms of paramilitary violence."
In London and Dublin, the Commons and the Dail met simultaneously to rush through anti-terrorist legislation aimed at ending the activities of the Real IRA, the group responsible for the Omagh attack. In the Dail, the Irish Prime Minister, Bertie Ahern, said the Real IRA had disgraced themselves, adding: "You cannot hope to take on the people of Ireland and win."
In another development yesterday, Mo Mowlam, Secretary of State for Northern Ireland, ordered the release of two Scots Guards, James Fisher and Mark Wright, who had been serving life sentences for the murder of a Catholic teenager in Belfast in 1992.
Her action was condemned by the family of the victim, and by Sinn Fein.
President Bill Clinton flies in to Belfast today for a short stay which will include a visit to Omagh with Mr Blair.
Pressure by the United States is believed to have been important in bringing about the flurry of activity which began with Mr Adams' statement on Tuesday.