I wouldn't swap picks with anyone in the top 6 unless we're getting a future 1st and 2nd rounder out of it. IMO any spot out of the top 8 would require multiple 1st rounders
By the way, I know the Browns don't have a head coach, and that means you may have to take this with a pinch of salt.
But it's worth noting that the Browns owner thinks he needs to show the fan base that he is serious about improving this team. He sent out an open letter to the fans after all of the top HC candidates took positions elsewhere, leaving the Browns as the only team currently without a HC. He also let the good quality coordinators he had in place (Norv Turner, Ray Horton) leave to other teams, showing no desire to keep them. Haslem is desperate to show the fans that he is wanting to improve the team. By coming out and saying that he wants to trade up for Manziel, he is hoping to achieve that. At the end of the day, he wants the Browns fans to come to games, to buy season tickets. I'm not saying that this makes the report true, but if any franchise in the NFL is going to do something like this at the moment, it's the Browns.
Anyway, whoever ends up as HC of the Browns may not have control over personnel decisions, so it may not matter what they think ultimately! That may be the sticking point as to why they haven't been able to find someone so far too...