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The tight relationship between Apple Inc. (NASDAQ: AAPL) and Hon Hai Precision Technologies, better known as Foxconn, appears to be fraying. The Taiwanese-based manufacturing giant is being pressured by another Taiwanese firm, Pegatron, that has been underbidding Foxconn in a successful effort to win business with Apple.

Pegatron has won contracts from Apple to build the iPhone 4S and the iPad mini. A report in the Financial Times this morning notes that Pegatron is seeking to increase its current workforce of 100,000 by 40%, still a far cry from Foxconn’s million or so workers.

The split between Apple and Foxconn apparently has less to do with the deaths and attempted suicides at Foxconn’s mainland China plants in 2010 than it does with cost. According to the FT, Foxconn tried to pass on to Apple wage increases for its Chinese workers and some factory relocation costs. Apple did not bite.

Foxconn has tried to diversify, but its success at doing so has been spotty. Apple has said that it would open at least one U.S. manufacturing in 2013. Foxconn employs about 3,000 U.S. workers at plants in Texas and California, and a natural assumption would be that Foxconn would get the additional work from Apple. That is not such an obvious conclusion now.