Bilbao, on Spain's north coast, is cooler and wetter than travel destinations to the south, making summer temperatures in Bilbao and the Basque country pleasant. Tourism peaks in August, so you may want to avoid that month.
April showers are quite frequent in Bilbao, with the rainfall peaking at around 12 inches for the month. There is considerably less rainfall in the summer months. Winter temperatures are quite moderate, but the weather can get quite grim during the off-season months.
Most tourists go to Bilbao for only a short visit to the Guggenheim, since it's a rather industrial city.
As usual for this part of Europe, May, June and September are the best times to visit. If you must go in August, reserve your lodging in advance, as this is the major vacation time in Europe. Also in the middle of August is the Semana Grande, a big festival in Bilbao.
The chart below shows the historic high and low temperatures in Bilbao, with the rainfall below that.
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Ibis Bilbao Centro (value)