Castro and Mexican President Enrique Pena Nieto are expected to discuss the issue when they meet in the eastern city of Merida on Friday, in their latest effort to relaunch ties strained under previous governments.
More than 1,500 Cubans reached Mexico in October alone, the latest arrivals in a recent surge that has "surprised" the authorities, Mario Madrazo, a senior official at the National Migration Institute, told AFP.
A higher number are leaving the communist-led island over fears that the US-Cuba diplomatic detente will lead to an end to the special treatment of Cuban migrants, who get automatic residency permits when they set foot in the United States.
Many are now choosing a long land route through several Latin American nations, rather than the risky trip across the shark-infested Florida Straits.
Cubans young and old, with or without children, have flooded the migration center in the town of Tapachula, in southern Chiapas state, after crossing a river from Guatemala a route taken by Central Americans for years.
Overall, 27,296 Cubans have reached the United States during the first nine months of the 2015 fiscal year, up 78 percent from the same period last year, according to the Pew Research Center, citing official figures.
Two-thirds of them, or 18,397, arrived through Laredo, a Texas city bordering the Mexican state of Tamaulipas 66 percent more than last year.
Mexican government figures show a similar trend, with nearly 6,500 Cubans taken to migration centers in the first nine months of this year, triple the number for the 12 months of 2014.
Many Cubans begin their trek after flying into Ecuador, the only Latin American country (aside from small Caribbean islands) that does not require them to have a visa.
Yordan Acosta's journey began in Quito on October 6. The 32-year-old veterinarian from Havana arrived in Tapachula on Tuesday, hoping to reach America to help his wife and two small children, who stayed back in Cuba.
For a month, Acosta and 11 of his Cuban friends traveled across Ecuador by car, passed through Colombia, took a boat to Panama, and rode on several buses in Costa Rica, Nicaragua, Honduras and Guatemala before reaching Mexico.
"There have been a lot of problems on the road. The police are allied with criminals, they ask you for a lot of money and they want to rob you," he told AFP by telephone, estimating that he spent more than $2,000 for his trip, including bribes.
Like others, Lil Susana Ponce, a 25-year-old mother of a seven-year-old girl, said she "just wants to improve my financial situation and help my family."
Ponce turned herself in to immigration authorities in Tapachula with the hope that they will issue a special permit to cross the Laredo border.
This letter of safe-passage, which is good for 30 days, is given to migrants whose country cannot identify them as citizens and does not demand their repatriation.
Only 10 percent of Cuban migrants are claimed by Havana, so there is a "minimal risk" for them to go to Mexican migration officials, Madrazo said.
'The sea is for fish'
The exodus of Cubans from their island is nothing new.
The US government's "wet foot, dry foot" policy allows Cubans who touch US soil to get permanent residency, while those caught at sea are returned to Cuba.
The fear that the US government may stop issuing automatic visas for Cubans has fueled the current Cuba emigration, though many are doing it by land now.
Only five percent of Cubans arrive in Mexico by boat, according to Mexican officials.
"Land is for people and the sea is for fish," Acosta quipped. "If a raft capsizes or is loses air, there's nothing you can do although you face more dangers by land."