Update 3.41pm: Micheál Martin has welcomed the appointment of Simon Coveney as Tánaiste.
"I want to congratulate Minister Coveney on his appointment which obviously has to be verified for the Dáil," he said.
"I'm glad that the Taoiseach is at last beginning to realise the enormous reservoir of talent and ability that resides in the county of Cork," he added.
Elsewhere, Heather Humphreys was made Minister for Business with Josepha Madigan being elevated to the role of Minister for Culture.
The changes came about following the resignation of Frances Fitzgerald as the Tánaiste.
Meanwhile, the Transport Minister said he is no doubt Simon Coveney will be "superb" in the role.
Minister Coveney is taking over the role after Frances Fitzgerald resigned.
Heather Humphreys will become Enterprise Minister while Josepha Madigan has been promoted to Culture Minister.
Minister Shane Ross welcomed the appointments.
"Simon Coveney will be a superb Tánaiste. I have no doubt whatsoever about that. He's a very worthy successor to Frances Fitzgerald and others," he said.
"I welcome the fact that Josepha Madigan has been appointed to the Cabinet. I think she'll be extremely effective and I look forward to working with her.
Update - 12.07pm: Foreign Affairs Minister Simon Coveney has been appointed by Taoiseach Leo Varadkar to the position of Tánaiste, succeeding Frances Fitzgerald who resigned earlier this week, writes Daniel McConnell of the Irish Examiner.
Normal proceedings in the Dáil were suspended to allow Mr Varadkar make the announcement which sees his main leadership rival elevated to the position of Tánaiste.
Minister Heather Humphreys has been moved to the Department of Business Enterprise and Innovation and she will now be replaced in her old department of Culture, Heritage and the Gaeltacht by first time TD Josepha Madigan, the Dublin Rathdown TD.
The elevated ministers will this afternoon will travel to Aras an Uachtarain to receive their seals of office.
In the Dáil, most Opposition leaders including Micheal Martin, Mary Lou McDonald, Brendan Howlin, Eamon Ryan and Richard Boyd Barrett wished the new appointees well in their posts.
However, Ms McDonald and Solidarity TD Ruth Coppinger did call into question the appointment of Ms Madigan to the Department of Culture given what they described were her anti-Traveller comments in previous times.
Ms Coppinger described her as engaging in “anti-traveller bigotry” adding it “said a lot” about this Government's attitude to minorities.
Announcing the new appointees, Mr Varadkar said: “Appointing the Minister for Foreign Affairs and Trade, Simon Coveney TD, as Tánaiste will enhance his role in the Brexit negotiations currently underway and will make it easier for him to coordinate the work of other departments with respect of Brexit.
“In appointing Minister Humphreys to the Department of Business, Enterprise and Innovation, for the first time in seven years the minister will be from outside the Dublin region, and also from the Border. This is particularly relevant in the context of Brexit and also the Government’s commitment to bring more jobs and good jobs to all regions of Ireland, especially rural Ireland. Heather’s experience as a minister and also her background in banking and finance makes her very suited to this role.
“Minister Madigan will build on the work of Minister Humphreys in the Department of Culture, Heritage and the Gaeltacht, and will drive forward the Government’s agenda for arts, culture and heritage, including the Creative Ireland programme, an ambitious ten year capital plan for the arts and multi-annual increases in funding for culture and heritage.”
The nominations were approved by the Dáil.
9.45am: Latest: Richard Bruton distances himself from Tánaiste job
Richard Bruton has distanced himself from the vacant role of Tánaiste.
The Taoiseach will announce Frances Fitzgerald's replacement later, with Simon Coveney and Heather Humphreys the favourites.
The role of Business Minister also has to be filled, so a Junior Minister or first time TD may be promoted.
Education Minister Richard Bruton's name has been in the mix for Tánaiste, and he was asked if congratulations are in order.
He said: "You'd be foolish to do that. I think the reality is that the Taoiseach will make his own choice, that's always the way.
"I think we're a bit of a gambling couuntry so there will always be odds being kicked about even on the darkest horses."
6.59am: Taoiseach under pressure to address gender imbalance as he prepares to name new Tánaiste
The Taoiseach Leo Varadkar is due to name the new Tánaiste later today.
It follows Tuesday's resignation of Frances Fitzgerald from the role.
He was Mr Varadkar's opponent in the battle to lead Fine Gael, but the Foreign Affairs Minister Simon Coveney was an early favourite to be announced as Tánaiste today.
However, also hotly tipped is Minister for Culture, Heritage and the Gaeltacht Heather Humphreys.
Further back in the field contenders include Finance Minister Paschal Donohoe and Housing Minister Eoghan Murphy.
The departure of Frances Fitzgerald from the Cabinet also leaves the Taoiseach with the Business, Enterprise and Innovation role to fill.
Given that he has been criticised for the gender imbalance in his Cabinet, there is speculation the next holder of that portfolio will also be female.