Lees-Galloway has been under pressure for granting residency despite Sroubek being found guilty of using a false passport and serving a prison sentence for drug-smuggling.
Speaking at the Labour Party conference this afternoon, Ardern said she spoke with Lees-Galloway this morning.
"We've absolutely agreed this needs to happen much sooner than three weeks."
Lees-Galloway has not discussed the detail of the claims, but a High Court judgement revealed that Sroubek had been back to Europe in 2009, undercutting any belief that Sroubek's life might be in danger if he were deported.
Lees-Galloway was not aware of the court judgement, and said it was not standard practice to have all court decisions relating to Sroubek handed to him.
Ardern defended him, saying ministers had to rely on the information put before them.
She said the investigation would look at why the court judgement was not passed on to the minister, but she denied she was throwing immigration officials under the bus.
"There's clear suggestion at this point that things have failed in this case. That's what we've asked the department to go back and work [through] as quickly as possible."
She publicly criticised the process for the first time.
"He had certain information in front of him. Now we have contradictory information in the public domain. Of course that's not good enough."
She said the focus was on Sroubek's case at the moment, but the investigation may uncover systemic issues.
"If that turns out to be the case, we would absolutely need to rectify that and get to the bottom of that."
The National Party has called for the axe to fall on Lees-Galloway, adding that the investigation is not needed and Sroubek should be deported.
"Despite the claims of fear of being sent back to the Czech Republic, he was voluntarily going back there and living in plain sight," National's immigration spokesman Michael Woodhouse said yesterday.
"One way or another, this fellow is not worthy of staying in New Zealand and he should be sent home."
Meanwhile, the Justice Ministry of the Czech Republic announced that it would begin extradition proceedings for Sroubek.
"Karel Sroubek was sentenced to four years and six months in prison in the Czech Republic. At the same time, the Czech Republic is prosecuted for further crimes. In 2013, an international search was announced," a translated version of the Ministry of Justice statement says.
The ministry had been waiting for deportation, but would now ask for extradition because Sroubek's deportation liability had been cancelled.
"The Ministry of Justice will do so in the near future."
Justice Minister Andrew Little told Parliament on Thursday Czech officials had not yet requested Sroubek's extradition.
A Parole Board spokesperson said that Interpol had provided the board with paperwork signed by a Czech judge that outlined offences for an extradition.
"Whether that request was advanced by the Czech government is not a matter for the New Zealand Parole Board, and has no bearing on its risk assessments."
Sroubek came to New Zealand on a false passport from the Czech Republic in 2003, but was unmasked in 2009 when New Zealand police were given details of his identity and an international arrest warrant for minor charges.
In 2011 he was found guilty of using a false passport, but was discharged without conviction on grounds that he would be in danger from corrupt Czech authorities if he were deported.
He was previously acquitted of committing aggravated robbery with members of the Hells Angels, but in 2016 was jailed for five years and nine months for importing 5kg of MDMA.
