The 35th UK-France summit came as Britain tries to strengthen bilateral ties before leaving the European Union in March 2019.
May said she was 'honoured' by the loan of the Bayeux Tapestry and Macron said he hoped it would open "a new chapter" for cultural and scientific co-operation.
"We are somehow making a new tapestry together," Macron said, rejecting any suggestion that it was intended to remind Britain of a victorious France.
Experts give one in the eye to Bayeux Tapestry loan offer
The loan has been met with opposition by French experts who have warned about the delicate state of the 70-metre (230-foot) long embroidery.
The tapestry, which shows how the English King Harold being killed by an arrow in the eye at the Battle of Hastings in 1066, is believed to have been created within years of the battle.
The UK-France summit came as Britain tries to strengthen bilateral ties before leaving the European Union in March 2019.
The French flag was raised over the military base and the French national anthem played out for the arrival of May and Macron, who were greeted by a guard honour and a flypast of two planes.
They earlier visited the Michelin-star Royal Oak pub in May's Maidenhead constituency, where they had a 'convivial' lunch together according to a British source.
While Brexit was not scheduled for formal discussion, Macron said Britain's decision to leave the European Union would lead to some "short-term uncertainties".
Macron also said that Britain could have "no differentiated access to financial services" if it decides to leave the European single market.
Otherwise, Britain would have to pay into the EU budget.
"There shall be no hypocrisy in this respect... otherwise we would destroy the single market," he said.
The French president stressed that the new border agreement was a bilateral accord, separate from any EU divorce deal, aimed at managing the sensitive of immigration as hundreds of people continue to camp out in Calais.
The two countries currently abide by the 15-year-old Treaty of Le Touquet, which permits immigration checks within each other's borders.
The new treaty signed at Thursday's summit was intended to complement the 2003 deal, according to French officials.
In addition to more funds from Britain, the two sides agreed to "improved treatment" of child refugees which Macron said would see their paperwork processed within 25 days.
The British prime minister also committed to sending Royal Air Force (RAF) helicopters to a key French counter-terrorism operation in Mali.
EU says 'hearts are still open' to Brexit reversal
The deployment of three RAF Chinook helicopters to provide logistic support to French troops tackling jihadis across Africa's Sahel region is part of broader counter-terrorism and military efforts there by the UN, the EU and the African Union.
It is seen as particularly significant as France is lacking in such capabilities and Britain's commitment could mark the start of a longer-term deployment in the region.
"Since the last UK-France summit we have seen appalling and cowardly terrorist attacks in both our countries, in which British and French citizens have died side-by-side," May said.
France in turn has agreed to commit troops to the British-led North Atlantic Treaty Organisation (NATO) battlegroup in Estonia in 2019.
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