Business Secretary Sajid Javid has rejected a call from a former head of the Army to send ground troops to the Middle East to fight Islamic State.
Writing in the Mail on Sunday, Lord Richard Dannatt says it may be time to think the "previously unthinkable" to solve the "evil problem" of IS.
He says all Britain has done so far is "fiddle while a modern Rome burns".
But Mr Javid told the BBC's Andrew Marr Show the war has to be won by those most affected - Iraqis and Syrians.
In recent days, Islamic State militants seized the ancient Syrian city of Palmyra, locked its museum and raised their flag on the ancient castle overlooking Roman ruins.
IS has previously demolished ancient sites that pre-date Islam in what they see as a war against "false idols".
Lord Dannatt says the capture and "likely destruction" of the 2,000-year-old Roman colonnades constitutes a "potential cultural crime on a gigantic scale".
He is calling for a public and political debate to begin "immediately" so the arguments for and against sending in British and US ground troops can be aired.
"Military history tells us that while air strikes can change the terms of battle, they cannot bring a decision," he writes in his article.
"People live on the ground, in towns, villages and houses - and on the ground is where the evil problem that is IS must be solved. To date, all Britain has done is 'fiddle while a modern Rome burns'.
"We have now reached a point when we must think the previously unthinkable and consider that British troops, acting as part of an international coalition, may be required to mount a ground campaign in Iraq and Syria."
He added that he was "no gung-ho general" who says "just send the boys in and don't worry about the body-bags".
IS was a "lethal and uncompromising enemy" and the UK could no longer afford to rule out "boots on the ground", he wrote.
He suggested sending in 5,000 troops with attack helicopters, artillery, mortars, reconnaissance and surveillance assets.
IS operated in fully-formed units and used conventional tactics, thereby presenting targets for international military forces to strike, he wrote.
Conservative minister Sajid Javid said there was a need for troops on the ground - but not British troops.
The government needed to see what more could be done in terms of training, intelligence and equipment, he said.
"But the answer to this is not going to be British troops on the ground," he told the BBC's Andrew Marr.
"This has to be won by the people most affected - it's the Iraqis, the Syrians - and we've got to provide them help with our allies as we're doing."
Conservative former defence secretary Liam Fox said boots on the ground would require the US to be the "prime mover".
He told Murnaghan on Sky News: "I think that is out of the question under the Obama presidency."
In his last year in office, Mr Obama would not want a war against IS to be his legacy, he added, suggesting it was time for "better collective security".
So far, British forces have conducted air operations over Iraq against IS targets, as part of an international strategy involving some 60 countries including the US.
The UK government is also providing £40m in aid to support people across Iraq who have fled from Islamic State.
Last year, MPs voted against the coalition government's plans for military action in Syria.