The Barcelona running is significant because Pirelli only has one more session with its 2024 tyres – after the British Grand Prix at Silverstone with Red Bull among those taking part – before the FIA makes a final call on whether to push ahead with the original plan to run without blankets next season.
The rules currently say that blankets will be retained for 2024, and the decision on whether to change that and abandon them has to be made by 31 July.
Several drivers who tested early prototypes of the blanket-free tyres in the winter were not happy with them, with Lewis Hamilton branding them "dangerous".
However, a test after the Bahrain GP showed more promise, and Pirelli believes that next week's running in Spain will represent a further step.
An early take on the future construction will be used from this year's British GP after it was tested by drivers in Barcelona on Friday.
The FIA has already approved blanket-free full wet tyres, which were used for the first time by Sergio Perez and the Haas drivers during the Monaco GP.
"This is quite important for us," said Pirelli's chief engineer Simone Berra when asked by Motorsport.com about the Barcelona running.
"We are testing most of the compounds, and we have many variants. We are trying to test different constructions to move forward, because the target is no blankets for 2024.
"So we need to have a construction that obviously needs to have good integrity, high-level integrity, but needs to work really well when the pressure is low at the start of the run, and help the compound to get up to temperature and be in the working range.
"So we are continuing to develop new structures. And then we will start to focus on the new compound range with no blankets usage. So I think it would be a good test for us.
"Barcelona we know is quite representative, because we have high-speed sections. And it's a normal circuit, a more complete circuit compared to Bahrain, when we tested the beginning of the season."
Although not fully representative, Berra believes that the Bahrain testing helped to point Pirelli in the right direction, and that the tyres it will run in Spain next week will meet the requirements.
"I think we already obtained something interesting," he said. "Obviously, Bahrain is not really significant for all the circuits, because you have high track temperatures, and you have high traction demands and braking demands.
"So we need to test the new structures on different circuits where you have high lateral loads, and higher severity, and probably lower temperatures compared to Bahrain, just to see that we are working in a good direction.
"When we tested in Bahrain, we had already some good options which worked properly, out of the box. But there's still something to fine-tune, to finalise, and it's a new proposal.
"So that's the aim of the test. We are working to have a structure that can help in generating temperature on the tyres, to allow the compounds to work properly. That means that the structure is less stiff, and can increase the stiffness with increasing pressure."
Berra is confident that Pirelli will ultimately get FIA approval to abandon blankets in 2024.
The test will also see the Mercedes track debut of the team's reserve driver Mick Schumacher, who will drive the W14 on Wednesday after George Russell runs the previous day.
