A trade deal would not only safeguard trade but also buttress peace in British-ruled Northern Ireland, though some disruption is nearly certain at the busiest EU-UK border points.
Merkel said some European Union member states were losing patience.
Fishing rights as well as the governance of any deal and the "level playing field" have been the main stumbling blocks preventing the two sides from reaching a deal so far.
They said that while that may have worked for Britain in previous Brexit negotiations it will not work this time around.
Face-to-face talks are continuing in London in what is seen as a crucial week if an agreement is to be reached.
Coveney agreed that fishing is an outstanding issue but accused the United Kingdom of creating the perception "that they own all fish that are caught in their waters, as opposed to it being a shared stock, in many cases".
A failure to strike an agreement will see Britain and Brussels forced to trade on basic World Trade Organisation terms from January 1 which would see tariffs imposed on goods.
British Cabinet Office Minister Michael Gove leaves a coffee shop as an anti-Brexit protester stands outside, London, UK, November 30, 2020. We don't need an agreement at any price.
"The EU still wants to take the lion's share of the fishing in our waters - which is just not fair given that we are leaving the EU", Gove told Sky. "A deal is in everyone's interest", she said. There's not much time left'.
"I've been saying for the last number of weeks that this is the key week, and we're staying the same again this week", Coveney told Morning Ireland.
"We, Britain and the member states of the European Union, are countries that are based on the same values, and it would not be a good example for the world if we didn't in the end manage to craft an agreement".
Mrs Merkel's comments came as Lord Hague urged Mr Johnson to personally take charge of the negotiations to secure a deal.
'This will help us tackle challenges quickly and decisively, and give us increased information which will make us safer and more secure'.
'Otherwise, it is still possible that January 1 will come round with no framework for cooperation and trade in place at all'.
Simon Coveney says they're used to last minute deadlines in these talks.
He added: 'For national leaders not to assert themselves and force a solution later this week - if none is forthcoming from the talks - would be a failure of statesmanship'.
Britain and the European Union warned each other on Monday that time was running out to reach a Brexit trade deal, with big differences still to be bridged on state aid, enforcement and fishing.