STEM Readiness Act of 2013 - Amends the Elementary and Secondary Education Act of 1965 to direct the Secretary of Education to award competitive merit-based scholarships to students who are pursuing bachelor's degrees in science, technology, engineering, and mathematics (STEM) with concurrent certification as kindergarten, elementary, and secondary school teachers.

Requires scholarship recipients to work for at least five academic years as a full-time STEM teacher at a public or private kindergarten or elementary or secondary school during the seven-year period beginning within one year after they complete their studies.

Awards scholarships for one academic year of study at a time, but makes them renewable on an annual basis if their recipients meet certain measures of academic progress.

Allows scholarship recipients to enter into agreements with the Secretary that provide them with a bonus in exchange for performing their service in a high-need local educational agency for a period equivalent to the period for which they receive the bonus.

Amends the Higher Education Act of 1965 to authorize the Secretary to award competitive matching grants to up to 50 institutions of higher education to establish, strengthen, and operate four-year undergraduate degree programs that enable students to concurrently: (1) earn a STEM bachelor's degree; and (2) be certified to teach kindergarten, elementary, or secondary school.

Awards those grants one fiscal year at a time, but makes them renewable on an annual basis for up to five years.