The Trello team announced in May 2015 that offline support is something they are "working on".
That said, it's been crawling through their features process since 2012, and they warn "it will be a long while", so don't hold your breath. You can use the above link to subscribe to this proposal, and/or vote in support of this proposal (at time of writing, it had 3,214 votes and was one of the most popular requests).
Here's the key information from that link:
Offline support in list Working on
Yes, we are working on this. We have a plan on what we need to do.
It's a big project and it will be a long while before we have
something to show you.
- Daniel Lew moved this card from Up next to Working on
27 May 2015 at 18:23
- Hamid Palo moved this card from Feature Ideas to Up next 17 Dec 2014 at 15:23
- Hamid Palo added this card to Future Features 15 Aug 2012 at 14:15