Gauck said "our hearts are wide open," but Germany faces difficulties housing refugees and providing schooling and other services amid budget constraints.
"Our absorption capacity is limited, even if it has not yet been decided where these limits lie," Gauck said at an intercultural service on Sunday evening.
With Germany expecting up to 1 million migrants by the end of the year, Gauck said the government will be required to "promote the construction of apartments and build schools, hire teachers and kindergarten staff, adjust the labor market and vocational training, teach the German language and German law -- and do all of that at the same time."
Germany is grappling with the influx of refugees, many from war-torn Syria, by speeding up the processing of asylum claims and establishing quotas for each of the country's 16 states after Bavaria was overwhelmed by refugees entering from Hungary and Austria.
The federal government on Thursday announced a series of measures that include providing states 670 euros per asylum seeker, per month. Leaders also agreed on tighter rules for those seeking asylum.
Gauck applauded volunteer efforts to help refugees, but the humanitarian response of people would need to be sustained "when the initial euphoria of giving has evaporated."
He voiced concern over "tensions between newcomers and established residents," while warning of "rightwing radical incendiaries and agitators" and the potential of Islamic fundamentalists to infiltrate Germany alongside refugees.
Extremist groups rise
Gauck's warning echoed comments by Germany's intelligence chief, who cautioned in a radio interview on Sunday of rising xenophobia and radicalization of right-wing groups responding the refugee influx.
"What we are seeing in connection with the refugee crisis is a mobilization on the street of right-wing extremists, but also of some left-wing extremists" against the right-wing groups, Germany's domestic spy chief Hans-Georg Maassen said.
He added that the intelligence service had noticed "a greater willingness to use violence" by far-right, left-wing and Islamist extremists.
The influx of refugees has led to a series of attacks and protests against refugees and their housing, as well as clashes between the far-right and leftist groups supporting refugees.
cw/bw (AFP, dpa)
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