Home Secretary Sajid Javid cut short a family Christmas break to return to the United Kingdom after declaring a "major incident" on Friday, as 39 migrants in small boats have arrived on the Kent coast since Christmas Day.
He spoke by phone on Sunday with French Interior Minister Christophe Castaner, who said they had agreed to "strengthen our actions to combat Channel crossings undertaken by certain irregular migrants on small boats, at peril of their lives".
He added: "Unfortunately there are no easy answers".
Since November, more than 220 people have attempted the journey.
Labour leader Jeremy Corbyn said in a tweet: "We have a duty to reach out the hand of humanity, support and friendship to people who are in danger and seeking a place of safety".
Mr Javid has come under growing pressure to act, and cut short a family holiday in South Africa to address the issue.
Mr Elphicke called for more patrol boats to be deployed in the Channel, and said: "For too long the Home Office has not been taking this seriously enough, and the crisis has continued to escalate".
Gavin Williamson, the defence secretary, offered Mr Javid the...
Six Iranian men were found on a Kent beach early on Sunday, in the latest successful crossing of the Channel from France by migrants in inflatable boats.
Lucy Moreton, a spokeswoman for the Immigration Services Union, said Border Force staff feel they do not have the resources needed to deal with the problem.
Monday's meeting to discuss further action will involve senior officials from Government departments and agencies, including Border Force and the National Crime Agency.
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The men made the trip in an inflatable boat with a rigid hull on December 30, landing in Kingsdown, according to the British Home Office.
"That is why I will continue to work closely with all partners to ensure we do even more to intercept these boats before they reach the United Kingdom, to tackle the organised crime gangs that facilitate many of these unsafe journeys, and to deter and protect those driven to make them".
Mr Javid declared the situation a "major incident" on Friday, appointing a "gold commander" to oversee the situation and give daily updates.
"There are a number of reasons behind the recent increase, many outside the UK's control".
But critics pointed out that the scale of the attempted crossings paled in comparison to the migrant crisis witnessed across the world, and that the issue has been ongoing.
A spokeswoman for the French Maritime Prefecture, Ingrid Parrot, said authorities were dealing with a "network" of organised people smugglers.
"Over the last three weeks, our French colleagues have disrupted numerous attempts, recovering at least 95 migrants including nine children, and arrested seven people caught attempting to facilitate these crossings".
Incidences of people trying to enter the United Kingdom over the last three weeks have involved "95 migrants including nine children", the National Crime Agency said on Saturday.
In November, French authorities said there had been almost double as many rescue attempts on the channel in 2018 as there had been the previous year, the Telegraph reported.
Security has been tightened at key locations in and around Calais where migrants have in the past congregated to try to get onto lorries for the final leg of the journey to the UK.