The key part of your question is in parenthesis: Will the Heat feel a need to bid against itself as the start of training camp nears? Because there doesn't appear to be any offers on the table for Lillard other than Miami's.
But what if the Heat does feel a sense of urgency to try to get a deal done before training camps around the league begin on Oct. 3? Unless another team becomes a serious contender in the Lillard sweepstakes, the desire to land Lillard before training camp is pretty much the only reason Miami would improve its offer to Portland in the coming weeks.
Are the Trail Blazers really willing to head into media day and begin camp with Lillard's unresolved trade request looming over the organization? Teams have done this before, but this situation is different than most because of Lillard's stature in Portland. The longer this goes, the messier it may get.
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