Behind-the-scenes negotiations have been going on between the UK Government and the DUP for months around Northern Ireland's place within the UK in post-Brexit arrangements.
The party is seeking further assurances from the UK about post-Brexit political and trading arrangements following the signing of the Windsor Framework.
The framework, negotiated by Prime Minister Rishi Sunak earlier this year and approved by Parliament, proposes reduced checks on goods travelling directly to Northern Ireland from Great Britain, to reduce trade barriers within the UK.
Speaking later, Northern Ireland Secretary Chris Heaton-Harris described the talks between the Government and the DUP as being at a "crucial stage".
He declined to give any indication of a timeline, telling the BBC he had learned from previous mistakes.
Mr Heaton-Harris said the Government is waiting for a response from the DUP and that he still feels optimistic, adding: "We are coming to a crucial point in the process where decisions have to be made."
Earlier Mr Baker urged hardliners to stop pressurising the DUP to stay out of Stormont, challenging them to consider what future they want.
"I think realistically where we are with the DUP is in the final stages of them letting us know what they need," Mr Baker told media in Belfast.
"But what I think I see is that a small number of really important opinion formers, who shape what the DUP does, are now really steering the life of Northern Ireland to a degree which is not really consistent with their status as unelected figures who express opinion and put pressure on the DUP.
"I would say to them and to everyone else, we now need to think really seriously about which futures are available and anyone as an opinion former who is now putting pressure on the DUP to stay out of government, they really need to ask themselves what they are trying to achieve."
Mr Baker said the EU will not re-enter negotiations, describing the Windsor Framework as "the best we could do ... our compromise".
Speaking in Belfast as a major investment conference comes to the city, Mr Baker urged that the goodwill is taken advantage of.
"We have got investors converging on Northern Ireland in a spirit of goodwill, wanting to invest. We have talked about how Northern Ireland is going to be in a unique place in all the world to take advantage of it," he said.
"So what I'd say to, for want of a better term, hardline unionist opinion formers, please think about the best interests of Northern Ireland. If you want Northern Ireland in the Union, it's going to need a government, so please give it that government."
Earlier DUP MP Gregory Campbell claimed Downing Street has been "repeatedly briefing" media that "substantial progress" has been made on talks with the DUP, "pressurising" his party to go back into Stormont.
Mr Campbell also said his party provided the Government with a paper, which he said they have been looking at for several months.
"The time has come for briefing designed to pressurise us to stop and delivery on what they have repeatedly promised to happen," he added.
Mr Baker responded: "Let's be absolutely clear here, I'm not coming out here today to try to brief against the DUP.
"We've got an investment summit coming up, and I want to see that be a success. That's why I've been allowed to come out and do media by Number 10.
"But it's just inevitable that you're going to ask me about executive formation, and it's just inevitable that that means we have got to talk about what the DUP are doing.
"I want the DUP to be happy and successful in government, but I'm afraid all the time they're out of government we're bound to be talking about it.
"I'm very fond of Gregory, but they've talked about megaphone diplomacy and all the rest of it, I'm answering your questions, I'm happy to answer your questions but it's in the DUP's hands if they want Northern Ireland to flourish.
"They just need to go and govern Northern Ireland, and we're very, very happy but at the moment I have to tell you, we're waiting on the last bit of information from the DUP about what they want and when they get it we will strain every sinew to give it to them because we want them back."
Mr Heaton-Harris was later asked who he thought Mr Baker was referring to, and responded: "I don't know, I haven't seen his transcript or indeed seen him since. The DUP has very important opinion formers, and indeed lots of people who built the party."
Questioned further, the minister said: "He's (Mr Baker) bang on, he's right."
The loyalist Jamie Bryson, NI director for Centre for the Union, described Mr Baker’s comments as “yet another effort to bully unionists, and tell those of us opposed to the Protocol to sit down, shut up and swallow the misleading claims on the Framework”.
He added: “On the footing that I presume I am one of the people Baker refers to as a unionist hardliner– and I wear that badge with pride – let me send a clear message back to him.
“No self respecting unionist or loyalist will ever compromise, weaken or give in on the Union. Returning to Stormont requires implementing the subjugation of the Union, and that is something that no self respecting unionist will ever do.”