The monthlong government shutdown is forcing some companies to seek alternate routes to go public because the main markets regulator is unable to green-light IPOs.

With Democrats and Republicans no closer on Monday to ending the impasse, biotechnology companies Gossamer Bio Inc. and TCR2 Therapeutics Inc. have been exploring a little-used workaround that would let them begin trading without the usual U.S. Securities and Exchange Commission signoff, according to people familiar with the matter.