He also applauded an "apparent u-turn" by Theresa May on the rights of EU citizens living in the UK.
Jean-Claude Juncker confirmed this morning that Brexit talks can move on to the second phase after both parties achieved "sufficient progress".
Brexit deal: Here's how British politicians reacted
"The Prime Minister has done what is necessary to get the talks on to the next stage - but we should have got to this point many months ago," Khan said.
"The government need to now accelerate progress - we simply cannot afford further delays. It is clear from the many conversations I have with business leaders that they need to know the details of the interim deal as early as possible in order to avoid having to put contingency plans into place.
"I welcome the apparent u-turn from Theresa May on the rights of EU citizens living in the UK, and British citizens living in the EU. I will pore over the details of this over the coming days because there needs to be clear and unambiguous reassurance to the three million EU citizens in Britain – one million of whom are Londoners – that they can stay, and that they’ll have automatic full rights."
However, he added: "It is extremely disappointing that the Prime Minister is still insisting that Britain will leave the single market and customs union. Staying in both is without a doubt the best long-term solution for London and Britain.
"Despite the progress today, it looks increasingly unlikely that we will get a deal that works in London's best interests and protects jobs and growth across Britain."