Redknapp was considered the clear favourite for the post as soon as the previous incumbent Fabio Capello resigned in February, but the FA caught many people by surprise on Sunday as they approached West Brom boss Hodgson to take over the Three Lions instead.
Redknapp insisted that he is not disappointed to have been overlooked, and believes Hodgson will do a fine job with the national team.
"If Roy takes the job I wish him all the best, he's a great guy," Redknapp told Sky Sports. "I've got a great job here, working with some fantastic players. Great result yesterday so I've got no problems whatsoever. I'm very lucky to be working at such a good club as Tottenham. Very fortunate."
He added: "I like Roy, he's a top man - a good guy and I'm sure he'll be a big success. I still think England has some fantastic players, some great young players coming through and some experienced players too. I look forward to watching the European Championships."
When asked whether the whole process had left Redknapp annoyed by the FA, the former West Ham and Portsmouth manager denied that was the case - although he did admit that perhaps the whole thing dragged out too long and could have been handled slightly better.
"I don't get like that, I don't hold grudges," he said. "I'm very fortunate. Lots of people would give anything to have the job I've got here, I'm so lucky to be working here with big players and in big games. I don't feel like that about it. I don't hold regrets.
"I suppose it has dragged on, but it's no problem. I just get on with my job here; we've got the end of the season to come, we've had a great season so far and we've got three games left now to try and get a Champions League spot - that's where my focus is. That hasn't ever changed."
With the England job seemingly no longer a possibility, Redknapp is set to continue his duties at White Hart Lane. With his current deal set to expire in just over 12 months' time a new contract could be in the offing, although he insisted that there had been no discussions about that as yet.
"I don't know anything about that," Redknapp said. "That's up to the chairman. I haven't asked about that."