Horst Seehofer, the conservative premier of Bavaria, said he did not share Merkel's "We can do it" credo on accommodating the almost 1.1 million migrants and refugees who arrived in 2015.
Seehofer, who leads the Christian Social Union, the sister party to Merkel's conservative Christian Democrats, said, "with the best will, I cannot make it mine. The situation is too problematic."
Speaking after a meeting with the Bavarian government in Tegernsee, he added that the solutions to date were "too inadequate."
Stressing he had no wish to start a quarrel with Merkel's party, Seehofer said it was important to look "reality" in the face.
An axe rampage, a shooting spree, a knife attack and a suicide bombing in the span of a week stunned Germany, leaving 13 people dead, including three assailants, and dozens wounded.
Three of the four attackers were asylum seekers, and two of the assaults were claimed by the Islamic State group.
On Thursday Merkel said that she would not allow jihadists to keep her government from being guided by reason and compassion.
"Despite the great unease these events inspire, fear can't be the guide for political decisions," she said.
"It is my deep conviction that we cannot let our way of life be destroyed," he added.
After the Bavaria attacks, Seehofer initially called into question the principle that asylum seekers should never be sent back to war zones. He later backtracked, citing international law.
However, he insisted previously: "We must seriously consider how such people should be treated if they violate the law or can be considered a danger."
On Saturday he cited the security situation in France, Germany and specifically Bavaria, saying there was an urgent need "to take action."
"That's why, here in Germany, we still have some way to go to improve in all areas," he said.