Britain's policy of tackling small boat crossings is set to face pressures from both the ruling Conservative and opposition Labour parties over the incident, which left at least six dead.
Home Office Secretary Suella Braverman offered her condolences.
Bridget Phillipson, Shadow Secretary of State for Education and Labour MP for Houghton and Sunderland South, said such incidents were "absolutely tragic."
The incidents in the English Channel “demonstrate why we need much tougher action to crack down on criminal gangs that are exploiting people, putting them in harm's way," she told BBC.
Phillipson described the Home Office as being "increasingly shambolic and completely incompetent.”
She pointed to the need for a "better, fairer system" to address a backlog of asylum applications "that is completely out of control."
Jake Berry, a backbench Conservative MP and former party chairman, said it is clear that "only radical changes can truly turn the tide."
"We must put a stop to the vile people smugglers who trade in human misery and whose actions continue to result in this loss of life," he wrote in the Sunday Express.
Recalling that all asylum seekers were moved from the Bibby Stockholm barge in less than a week, Berry also noted that the Rwanda deal still has not seen a single migrant sent to the African state.

On Friday, all asylum seekers were moved from the Bibby Stockholm barge after traces of Legionella bacteria were found in the on-board water system.

The volunteer-run charity Care4Calais said Saturday's incident was "appalling."
The “unnecessary loss of life affirms yet again the need for safe passage and a modern, sensible system of asylum in the UK," it said in a statement.
Meanwhile, conservative ministers are reportedly facing calls to resign after removing migrants from the barge.
A senior Tory figure told the i newspaper that Braverman "should be sacked" after all of the migrants on board the vessel were removed over bacteria concerns.
A Home Office insider said the issue was "embarrassing."
"What a complete & utter farce. As if having porous borders isn’t bad enough, we can’t even move 39 illegal immigrants onto a barge properly," Scott Benton, an independent lawmaker who is a former Conservative MP, wrote on X, formerly known as Twitter.
According to data from the Home Office, a total of 1,608 migrants in 30 small boats were detected crossing the English Channel from Aug. 10-12.
Prime Minister Rishi Sunak listed tackling small boat crossings as one of his five priorities after over 45,000 migrants arrived in the country by crossing the English Channel last year.