The power-sharing government is a cornerstone of Northern Ireland's peace process and has been dominated by unionists.
A collapse of trust led to the withdrawal of the Irish republican Sinn Fein party, which triggered a March 2 snap election in which the conservative, pro-British Democratic Unionist Party (DUP) finished only narrowly ahead of them.
After three months of negotiations, agreement has yet to be reached between the two main parties, who between them hold 55 seats in the 90-seat local assembly.
Without a deal, the province would likely revert to direct rule from London as a second snap election would run the risk of further polarisation of the divided electorate.
"I continue to believe that a deal remains achievable, and if agreement is reached, I will bring forward legislation to enable an executive to be formed possibly as early as this week, but time is short," said Brokenshire.
The 1998 Good Friday Agreement forged a peace process that ended three decades of violent conflict in the province, with responsibility for health, education, justice and the province's economy handed over to the local government.
In London, British Prime Minister Theresa May's party struck a deal with the DUP last week that will afford her Conservatives a slim parliamentary majority.
Northern Ireland will receive an extra 1 billion pounds (1.1 billion euros, $1.3 billion) from the UK coffers over two years, in return for the DUP backing the British government in confidence votes, passing budgets and Brexit legislation.
Sinn Fein wants legislation on the Irish language including its inclusion in areas such as broadcasts and street and road signs, same-sex marriage.
It has also demanded that DUP leader Arlene Foster step aside as first minister pending an investigation into a botched green energy scheme her party had championed.
Sinn Fein negotiator Conor Murphy on Monday accused the DUP of intransigence and a "lack of urgency" and said that its "anti-equality approach has been emboldened by their alliance with a Tory government.
"The DUP opposes rights for Irish speakers, it opposes rights for ethnic minorities and for women," he added.
"The political institutions must be built on equality so that a rights-based society can emerge."
The DUP believe Sinn Fein have politicised the Irish language, do not back gay marriage and insist Foster will remain in charge.
Foster on Monday said that "Sinn Fein have a shopping list, a shopping list that seems to get longer every time we meet with them.
"The onus is really on Sinn Fein now as to whether they want to continue with this political grandstanding or whether they want to get back to doing the job that we need to do," she added.
Without a functioning Northern Ireland executive, local politicians will not determine how the money is spent. Civil servants have been running Northern Ireland's departments in the absence of ministers.
Brokenshire warned he may need to give more authority over expenditure to civil servants if no deal is reached.