Border Force officials were called on Sunday morning after the men were found on a beach in Kent with a rigid-hulled inflatable boat.
A witness who works at the Zetland Arms pub, which overlooks the beach, said they saw authorities leading a small group of men from the shore to a Ford Transit van at around 9.30am.
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".
Immigration Minister Caroline Nokes said Javid was leaving his family holiday early to address the increase in crossings.
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Defence Secretary Gavin Williamson has said the UK's armed forces "stand ready" if needed to prevent further crossings.
Compared to other influxes of migrants and refugees, the numbers trying to cross the Channel recently are tiny.
The Home Office did not give information on how this would be done.
But Mr Javid faced accusations of over-egging the scale of the problem, with shadow home secretary Diane Abbott telling the Guardian: "There's no question that with Brexit, and also with the approach of the meaningful vote in January, people are being whipped up about migration issues, because the Government thinks this is the best way of frightening people to vote for their deal".
"It is vital we strike a balance between protecting them and protecting our borders, ensuring we do not encourage more people to make this unsafe journey".
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Officers attended to assist Border Force officials who are dealing with the case.
In another recent incident, eight Iranian men were spotted in a small boat near the busy port of Dover on Friday and brought ashore for medical assessments and immigration interviews.
Since November, more than 220 people have attempted the journey. Several arrests of suspected people smugglers have been made.
Detention Action said the reason for the crisis was the Government's failure to create safe and legal routes for people to claim asylum in Britain.
Chris Matcham, who watched the events unfold, said: "I was driving towards New Romney this morning with my sister when a police auto came haring past us along the road".
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"Instead of laying the blame exclusively at the hands of criminal gangs, Sajid Javid must step up and provide safe passage for those in distress if he wants to avoid further human tragedy".