Earlier in the day, the Ministry of Defence (MoD) said fighting to the north-west of Kyiv continued, with the bulk of Russian ground forces about 25 kilometres from its centre.
Meanwhile, the Ukrainian government said a mosque in the city of Mariupol had been shelled by the Russian military.
The mosque was said to be sheltering more than 80 people. There were no immediate reports of casualties.
Mr Johnson and Mr Martin spoke about the crisis in Ukraine and the Northern Ireland Protocol before watching Ireland beat England in the Six Nations at Twickenham.
A Downing Street spokesperson said both leaders discussed their “deep concern around the intensification of hostilities near Kyiv” and “condemned the attacks by Russian forces on civilians”.
“The Prime Minister said this was a critical moment for Ukraine and for wider European and international security,” the spokesperson added.
“They agreed on the vital importance of continued unity in the face of Russian aggression, including through humanitarian relief, defensive military support to the Ukrainian government and further sanctions to target the Putin regime.”
Pleased to meet Prime Minister @BorisJohnson before the Ireland v England game.
We discussed brutal invasion of #Ukraine by Russia and the unfolding humanitarian crisis.
Welcomed the close collaboration between the EU, U.K. and our partners to hold Russia to account. pic.twitter.com/hDmzTfeIkL
— Micheál Martin (@MichealMartinTD) March 12, 2022
Mr Johnson was said to have told Mr Martin he hoped the “same spirit of co-operation” the UK and EU had shared during the Ukraine crisis could be applied to protocol discussions.
A spokesperson for the Taoiseach’s office said the pair discussed “the brutal Russian invasion of Ukraine and the profound global security, economic and humanitarian consequences”.
They welcomed the “close collaboration” between the EU, UK and other partners to hold Russia to account and to address the humanitarian needs of the Ukrainian people, the spokesperson said.
The crisis in Ukraine and the protocol were expected to feature high on the agenda as the leaders met this weekend.
The Irish premier is visiting the UK as part of a programme of events to mark St Patrick’s Day.
Prior to the bilateral meeting on Saturday, Mr Martin said he was not there to tell the Prime Minister what to do with regard to refugees.
Asked if he would encourage Mr Johnson and the British Government to take in more people fleeing the war in Ukraine, he told reporters at the Embassy of Ireland: “Well to be fair I’m not here to tell Boris Johnson what to do.
“I have to acknowledge the leadership of the UK Government in terms of the sanctions it has imposed on Russia.
“And also the support it has given to Ukraine and indeed to many of the eastern European countries of the European Union who value the support that they’ve received from the United Kingdom and who have said this.”
On Sunday, the Taoiseach will attend London’s St Patrick’s Day festival and parade, which this year will celebrate “community” and key workers during the Covid-19 pandemic.