Coordination between heat and electricity markets is essential to achieve a cost-effective and efficient operation of the overall energy system. In the current sequential market practice, the heat market is cleared before the electricity market, and has no insight into the impacts of heat dispatch on the electricity market. While preserving this sequential practice, this paper introduces an ``electricity-aware'' bid format for the coordination of heat and electricity systems. This novel market mechanism defines heat bids conditionally on day-ahead electricity prices. Prior to clearing heat and electricity markets, the proposed bid selection mechanism selects the ``valid'' bids which minimize the operating cost of the heat system, while anticipating heat and electricity market clearing.} This mechanism is modeled as a trilevel optimization problem, which we recast as a mixed-integer linear program using a lexicographic function. We use a realistic case study based on the Danish electricity and heat system, and show that the proposed bid selection mechanism yields a 4.5\% reduction in total operating cost of heat and electricity systems compared to the existing market-clearing procedure, while reducing the financial losses of combined heat and power plants and heat pumps due to invalid bids by up to 20.3 million euros.