LONDON (Reuters) - British Brexit minister David Davis met the EU's chief negotiator Michel Barnier on Monday, the first meeting of several that Britain hopes will "lay the ground for a constructive dialogue".

His ministry, the Department for Exiting the European Union, said he would hold meetings in Brussels and Strasbourg this week with leading figures not to "seek to open negotiations" but to prepare the way for a "smooth and orderly exit" from the bloc.

"I want to begin the work of ensuring we have positive, strong, and productive relationship with our closest neighbors," Davis is expected to tell officials, his department said in a statement.

"I am confident that by working together we will be able to secure a deal that works in the mutual interests of the UK and the rest of the European Union."

Barnier, a French conservative former minister and EU commissioner, said on Twitter: "This morning courtesy visit from @daviddavismp at his request. No negotiation without notification. My work is now focused on EU27."

British Prime Minister Theresa May has said she will trigger Article 50 of the EU's Lisbon Treaty to start the formal divorce talks by the end of March, and EU officials have said there can be no pre-negotiations on their future ties before then.

May and her government are hoping to negotiate a "bespoke" deal with the EU - one that will allow Britain to control immigration from the bloc but also protect trade.

But EU officials have ruled out allowing Britain to "cherry pick" the elements of the union that it wants with none of the responsibilities, such as honoring freedom of movement.

(Reporting by Elizabeth Piper and Francesco Guarascio in Brussels; Editing by Toby Chopra)