Renting a car and driving from Girona to Bilao

Before I make some suggestions, may I ask what time of the year you will be making this trip? Some roads through the Pyrenees may still be snowed in, depending on timing.

Carlos.. early September this year.

Thank you, then that gives you some options, you won't have to worry about snow lol! I'd recommend these potential routes:

1. Along the Ebro River: goes through the heart of the region of Aragon and the southern Basque country:

Start Girona mid-day
-drive to the market town of Vic (have lunch),
-continue to
Zaragoza (2 nights) - see its Moorish Palace, Roman Ruins, and large Basilica
-continue west to
Olite (1 night) - stay at the castle Parador
-drive on the A-1/A-10 stop at Urbasa-Andia nature park
Vitoria-Gasteiz (2 nights) - stay at the hilltop Casco Viejo
-turn north
End Bilbao

or

2. Across the Pyrenees: this goes over the "rooftop of Spain" so to speak, dotted with ancient monasteries and castles:

Start Girona early-morning
- drive westwards on the N-260
-stop Besalú for breakfast
-stop at Queralbs take rack-railway up to Val de Núria for a few hrs
-continue on the N-260 until
Vielha (2 nights) - day trip to Vall de Boi (UNESCO Site)
-continue on N-260 to
Jaca (2 nights) - day trip to Loarre Castle
-drive west stop at Sos del Rey Católico
-continue to
Pamplona (1 night) - day trip to Olite Castle
-turn north-west
End Bilbao

I personally would go with option 2, especially considering the weather may be a bit hot with option 1, along the central Ebro area, where Zaragoza is.

Yeah, "a bit hot" might be an understatement considering what Zaragoza was like for me.

I'm unclear on whether 5 days is the total length of time between the end of your bike tour and your flight out of Bilbao or you have already set aside an unspecified number of days for the Basque Country.

It's time-consuming to get to San Sebastian and Bilbao, so I would want to see them if flying out of Bilbao.