Mr McConnell and wife Bridget addressed the media
Mr McConnell will now face an affirmatory ballot of fellow MSPs and the party's ruling body, before being formally proposed as the new first minister.
Earlier on Tuesday, Mr McConnell confessed to cheating on his wife seven years ago, in an attempt to see off rumours about his private life.
He will now seek formal approval as the successor to Henry McLeish, who resigned last week after revelations about expenses irregularities.
When nominations closed at 1700 GMT on Tuesday, the education minister was the only candidate seeking nomination.
The only other contender, the left-wing backbencher John McAllion, failed to receive the minimum number of seven nominations.
What had been expected to be a three-week-long contest for the Labour leadership is expected to be over in just days, with Labour's MSPs and the party's executive set to approve the nomination.
In a statement Scottish Labour said: "Jack McConnell received 33 nominations and is therefore the only valid candidate.
"Jack McConnell's name will now go forward to an affirmative ballot of the Scottish executive of the Labour Party and the Labour MSPs.
"This meeting will take place in the next few days."
Opposition leaders
Labour said it expected Mr McConnell would be confirmed as Scottish First Minister some time next week during a sitting of the Scottish Parliament.
It remains to be seen if Mr McConnell will face a challenge from opposition leaders or other MSPs in the parliament.
On Tuesday morning, Mr McConnell, confirmed newpaper reports that he had an extra-marital affair seven years ago.
Accompanied by his wife Bridget, he said he wanted to clear the air about the matter and avoid media intrusion into his private life.
His wife said: "It goes without saying that he hurt everyone involved."
However, she told reporters that the couple had rebuilt their marriage and their family.
On hearing of the closure of Labour nominations, Scottish Tory leader David McLetchie said: "Jack McConnell may have been given a walk over by his Labour colleagues but he can expect no such lenience from the Scottish Conservative Party.
"I look forward to tackling Mr McConnell in the chamber over his party's dismal failure on the issues that really matter to the people of Scotland."
Mr McLetchie said record of the Labour-Liberal Democrat Scottish Executive was "truly woeful"
Scottish National Party deputy leader Roseanna Cunningham described the leadership election as "a joke" with Mr McConnell the only candidate.