With a little more than a week left in Lightfoot's administration, there are questions about how the next administration will handle this crisis.
Lightfoot said more than 200 migrants a day are showing up in Chicago. Close to 300 migrants have camped on police station floors this week.
"We've got to make sure that we're doing everything we can to support the people that are here, but it doesn't make sense for people to continue to come to Chicago when we are literally out of space," Lightfoot said.
WATCH: Exploring Chicago migrant crisis
Exploring Chicago migrant crisis
Lightfoot said the city has run out of funds to aid the incoming migrants.
"We're going to keep demanding our fair share of resources from the federal government," Lightfoot said. "FEMA has got to step up and support Chicago, and they have not done that to the proportion to which I think we are entitled."
Lightfoot said she spoke to Johnson and his team last Sunday and urged him to name a point person to help manage this migrant crisis. Johnson said he has assembled a small team to work on a plan, but, so far, he has been short on details.
Johnson said the solution to the crisis will require collaboration with the state and the federal government for more resources.
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"We do need more resources from the federal government. We do. And yes, there has to be far more coordination, and, so, I've already assembled a very small team in the midst of my transition to work through a plan that brings people together," Johnson said.
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Rocio Velazquez Kato is with the Pilsen Neighbors Community Council.
Her nonprofit has been helping asylum seekers arriving in Chicago. On Friday, she put Johnson on the spot at a round-table event at Benito Juarez Community Academy.
"What we want to know is he is coming in with a plan on day one, and we have not seen that," Kato said.
Johnson refused to answer questions from the media on Thursday and earlier Friday morning about his plan.
"We didn't get any details on what exactly he's going to be doing to address this issue. We gave him very specific problems that we have, which is funding and coordination, but he didn't really tell us how he would solve funding and coordination issues," Kato said.
Alderman Byron Sigcho-Lopez said he has been working with the mayor-elect's transition team on a plan, but that Mayor Lori Lightfoot still has the ultimate word until she leaves office on May 15. Sigcho-Lopez said she failed to put together a plan months ago, when the migrants began arriving in Chicago.
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"We are working right now with the transition team quickly, and, unfortunately, put together a plan that was not created when it should have been months ago, when it was proposed to the current mayor," Sigcho-Lopez said. "From the city, from the current administration there is no blue print, there is no comprehensive plan."
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A meeting about opening another possible migrant shelter in the city at the old South Shore High School building was met with loud opposition on Thursday night. Many concerned citizens shouted down that proposal presented by city officials.
"We've heard from the community," Lightfoot said. "They have a lot of legitimate questions and concerns, and we're going to address to address those questions and concerns."
On Friday morning, Johnson said he wants to support asylum-seekers while also protecting Chicagoans who have been ignored.
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"I'm going to work very hard to make sure people seeking asylum are supported, but we also have to be very clear there are families that have not been supported for a really long time. Got to admit that," Johnson said. "I want to make sure that Black families who have been left out, and that the Brown families who want in, are not seen as a divided space."
Johnson also released a statement on the issue, again saying this needs to be a collaborative plan, which also involves engaging the community.
Groups helping migrants hope to get more details on a plan once Johnson takes office on May 15.
Johnson's full statement:
"The truth of the matter is that we have governors around the border who have demonstrated they are not open to collaborating, and that of course is a disappointment. I'm prepared to have conversations with everyone - particularly those who have used this as a weapon against cities - because asylum seekers are experiencing extremely difficult conditions, both emotionally and physically.
"What we will do day one is make sure there is a real effort to coordinate our resources at the federal, state, county and city levels. We are already in the process of developing the infrastructure for that type of coordination, but also for engaging in a real community process that allows for our response to not just meet the need, but include all stakeholders in solving what is ultimately a crisis that was engineered by outside forces.
"I'm going to work collaboratively with the people of Chicago and alderpersons who represent areas where we can be supportive for asylum seekers to make sure every community, and every person in it, is protected. Many communities have for too long been denied basic amenities and resources that could help strengthen them, but the stronger our communities are, the better position we are in to support both newcomers, and current residents and families."