Haier Group’s proposed $5.4 billion takeover of General Electric Co.’s appliance business would hand the Chinese manufacturer a substantial position in the U.S. market after a decade of trying.

The deal would vault the state-owned company past Electrolux AB and other rivals in the U.S. appliance market—currently led by Whirlpool Corp.—and help bolster profit margins that have been thinned by growing competition in its home market. Haier twice before tried to buy its way into American market. In 2005 it courted Maytag Corp....