The EU's call for comity between Algeria and Spain came after Algiers on June 8 announced a two decade-old "friendship treaty" with Madrid was put on hold.
The move was announced without stating reasons.
Observers say the suspension affects agreements on controlling the flow of migrants, as well as imposing a ban on Spanish imports.
Suspending the treaty could, according to observers, be extended to stop the flow of Algerian natural gas to Spain. But so far, both Algeria and Spain have indicated that no turn of the valve has been contemplated or officially threatened.
Algiers pulled its ambassador from Madrid in March after Spain approved neighboring Morocco's announced plan to grant Western Sahara autonomy but not full independence.
Algeria has, for years, backed the Polisario Front, a movement pushing for Western Sahara's right to be a nation.
By Spain backing Morocco's "autonomy" plan, Algiers said on Wednesday that Madrid had "given its full support to an illegal and illegitimate formula...advocated by the occupying power."
Meanwhile, adding to friction between Spain and Algeria, 113 undocumented migrants landed in Spain's Balearic Islands, which Madrid says is a favored transit destination for people seeking a place in the EU.
The cold front between Spain and Algeria has Brussels calling for reconsideration of Algiers' treaty suspension, saying the EU is "extremely concerned.."
EU foreign affairs spokesperson Nabila Massrali added "We would ask the Algerian party to take another look at that decision."