Coatbridge College head John Doyle was given the money after his institution merged with two other colleges.
Holyrood's public audit committee, which said the principal was paid "in excess of the guidelines", has passed its report to Police Scotland.
Mr Doyle told MSPs in October he had done nothing wrong and that his reputation had been unfairly "trashed".
In June, the Auditor General Caroline Gardner issued a highly-critical report of the severance deals paid out by Coatbridge College, which she said were overly-generous.
She claimed Mr Doyle and chairman of the former college - John Gray - had colluded in order to "achieve a certain outcome" by withholding important information from the remuneration committee.
The report by the Scottish Parliament's audit committee agreed with the auditor general.
Mr Doyle, who earned £116,000 a year by the end of his service, was given:
Six other college staff also benefited from the £850,000 pot of pay-off money.
Police Scotland said that officers had had discussions with Audit Scotland and the "matter is under review".
Following the report, the Scottish government said it would "carefully consider its [the report's] recommendations in the coming days".
Committee convener Paul Martin said: "There is a compelling moral argument for John Doyle to repay the tens of thousands of pounds extra he received from the college.
"The Scottish government provided more than £52m between 2011/12 and 2013/14 to support the college merger process and most of that money was used to fund voluntary severance schemes.
"It was not provided however to allow already highly-paid public servants to feather their own nests at the expense of their colleagues and of their students' education."
The report made key recommendations, including;
A Scottish government spokeswoman said the way in which "decisions involving hundreds of thousands of pounds of public money were made at Coatbridge College were completely unacceptable".
She added: "The events in question took place before April 2014 when colleges were reclassified by ONS and stronger financial controls came into effect."
By Glenn Campbell, BBC Scotland political correspondent
MSPs on Holyrood's public audit committee were unanimous.
Their report into bumper pay-offs for Coatbridge college bosses finds that the principal, John Doyle, was responsible for "serious failings in the governance of severance arrangements".
It says he colluded with the chairman, John Gray to secure "excessive" pay-offs for himself and six other senior managers by deliberately withholding information from those who signed off the deals.
The committee is urging Mr Doyle to pay back much of his £304,000 golden goodbye.
They say this request and their decision to copy their report to Police Scotland are without precedent.
Mr Doyle has previously refused to pay back any money arguing that he has "done nothing wrong". Mr Gray has also denied collusion.
The committee convenor, Paul Martin, says in these circumstances the charity regulator or the Scottish Funding Council should think creatively about how they could force a clawback.
The difficulty with these demands is that what MSPs find morally questionable may prove to be contractually correct.

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A former college principal should hand back some of his £304,000 severance pay, a committee of MSPs has concluded.
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