The 30-year-old, an integral part of Real Madrid's La Liga title win, said it would be foolish to underestimate the French and the English at the tournament which is being co-hosted by Poland and Ukraine.
"France and England are in that bunch of teams who are capable of going far," said Alonso, a member of both the EURO 2008 and 2010 FIFA World Cup™-winning Spain sides. "They are part of that group because of the players they have and their respective histories.
"They have this competitive instinct and even if they aren't talked about much with respect to this competition, they are more than able to attain their target (of winning it)," added Alonso, who has won over 90 caps since making his debut in 2003.
Alonso, who had a successful spell at Liverpool with the highlight the UEFA Champions League win in 2005, admitted that the season just past had left him and his team-mates exhausted but they had recovered since.
"The season in terms of club competitions was really intense," said Alonso. "We had to battle till the end in order to achieve our target, but we have had time to disconnect and because of that I feel fine personally."
Spain are due to fine tune their preparations for EURO with two more warm-up matches against Korea Republic in Berne, Switzerland, on Wednesday and China in Seville on Sunday after having already beat Serbia 2-0 at the weekend.
The Spanish will then fly to Poland on 5 June and play all their group games in Gdansk, against Italy on 10 June, then Ireland on 14 June and then finish their group games against 2008 quarter-finalists Croatia on 18 June.