There are very few circumstances where I think a manager has a legitimate reason to turn down Arsenal to stay at Leicester. This current however is one of them. If Rodgers goes to Arsenal and fails, he could be done at any decent sized club in this country. His reputation post Liverpool was hardly the greatest (we got some stick ourselves for appointing him) so to go to Arsenal and not deliver, which you've got to say is very possible given what a rebuild they need, it would be a big risk.
You've got a lot to lose leaving Leicester. What we are not is a massive club or a massive name. But when you weigh up what you do actually get at Leicester - it's a really good job. You get an excellent owner that will back you, you've got a fanbase that, while there aren't no expectations, they pale into insignificance compared to somewhere like Arsenal. You've got a very good squad already in place, loads of really good young players tied down for years - something that gives Leicester a lot of security going forwards as we've now got the power to get big money for them. This is a club going in the right direction for sure. It's a really nice set of circumstances for any manager, and I think Rodgers knows that.
Rodgers doesn't have to chuck everything away here for a rebuild needing Arsenal. What does he realistically expect to do there in the next 2/3 years that he's not going to do here this year? We've got every chance of making the top 4 this year. What are Arsenal going to do that's better than that? Does he think he could win them a title, or a European trophy?
If this was Man City or Liverpool, I'd totally get it. Arsenal? I think you're on a hiding to nothing.
Rodgers has got 20+ years left in management if he wants it. If he gets Leicester into the top 4, he could have a far better offer than Arsenal come in for him. There's a reason Arsenal ended up with Emery.
If Arsenal got their act together, put in a proper recruitment structure and invested in quality young players, I think the appeal becomes instantly significant. For now though? Poisoned chalice. He'd be making a mistake going there.