Education committee chair Robert Halfon has written to Education Secretary Damian Hinds, saying the move is needed to help tackle knife crime.
Handing councils greater powers to monitor exclusions could help promote early support to young people, keep them in mainstream education and deter them from knife crime, he writes.
Halfon highlights a "clear correlation between exclusions and knife crime", with young people not in mainstream education at greater risk of becoming involved in gangs and other criminal activity.
The powers would help local authorities track what happens to a disadvantaged young person with special needs as they pass through the education system - views expressed by former Ofsted chief Sir Michael Wilshaw recently.
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Halfon is also calling for improved guidance and better training to help teachers to support children at risk of being excluded and becoming involved in crime.
"Local authorities do not have a sufficiently strong role to play in scrutinising schools' approaches to exclusion in the first place," writes Halfon.
"We urge the department to consider giving local authorities more power to monitor exclusions."
Halfon is also calling on the government to speed up efforts to improve the quality of alternative provision (AP) for excluded pupils.
The Department for Education has allocated £4m through an alternative provision innovation fund to improve standards.
"However, we are concerned that the department is not moving urgently enough to improve the quality of AP," Halfon continues.
"In a number of local authority areas in England, no state-maintained AP place has been rated good or outstanding."
Halfon's letter follows an education committee evidence hearing on links between knife crime and school exclusions.
Among those giving evidence was Carlie Thomas, a senior caseworker at the charity St Giles Trust.
She said that early support for children affected by traumatic experiences can reduce the likelihood they will become involved in knife crime.
"There are a lot of tell-tale signs before they have got to the point of carrying a knife or being excluded from school," she said.
Thomas also backed training for teachers to help support children at risk of knife crime who are coping with trauma.
"I feel really sorry for our teachers and I feel really sorry for our education officials because they have not been given the relevant training or the right training," she told the hearing.
"They have been hung out to dry and expected to deal with these situations on their own. They do not have a clue what is going on."
The education committee also made available a letter from Hinds to Halfon ahead of their evidence session on school exclusions and knife crime.
Hinds' letter points out that in Scotland knife crime has risen despite exclusions falling.
However, he also concedes that children in alternative provision are more likely to be either a victim or perpetrator of crime or violence.
The education committee previously raised concerns about the quality of alternative provision and schools approaches to tackling knife crime in its July 2018 report Forgotten Children: Alternative Provision and the Scandal of Ever-increasing Exclusions
A government review into school exclusions, led by former children's minister Edward Timpson, is due to report its findings this year.
Earlier this month the government launched a consultation on a new public health duty requiring teachers, health workers and social workers to be required by law to raise concerns about children at risk of knife crime.