"Shots of undetermined origin" were fired during the attempt by around 600 immigrants during the night, radio Cadena Ser said on Thursday.
It said three were killed on the Spanish side of the border and three on the Moroccan side.
Local authorities in Ceuta did not immediately confirm the report, but said the head of the enclave's administration would issue a statement soon.
Spain's other North African enclave, Melilla, has seen numerous mass attempts by would-be immigrants to storm across the border fence into Morocco in recent weeks.
At least two have died in their attempts to scale the barrier at Melilla, which is also claimed by Morocco.
The issue of immigration will be high on the agenda at a bilateral Spanish-Moroccan summit on Thursday in the southern Spanish city of Seville.
Many of the immigrants are from sub-Saharan Africa and see Spain as the promised land just metres away after a trek across the desert.