In short, male to female transitions can participate under a set of conditions with respect to gender identity, hormonal treatment, and testosterone levels. Female to male transitions cannot participate if they are going through hormonal treatment.
WTA defines this in their Gender Participation Policy, most recently updated in December 2016:
The following guidelines set forth the eligibility of entry into and
participation of transgender players in WTA Tournaments:
A) A player
who transitions from male to female (also known as a trans female
(MTF)) is eligible to enter into and participate in WTA Tournaments
under the following conditions:
i. The player has declared that her
gender identity is female. The declaration cannot be changed, for
purposes of entry into any WTA Tournament, for a minimum of four (4)
ii. The player undergoes hormonal treatment for gender
transition and demonstrates that her total testosterone level in serum
has been below 10 nmol/L for at least twelve (12) months prior to her
first WTA Tournament (with the requirement for any longer period to be
based on a confidential, case-by-case evaluation, considering whether
or not twelve (12) months is a sufficient length of time to minimize
any advantage during competition).
iii. The player's total
testosterone level in serum must remain below 10 nmol/L throughout the
period of desired eligibility to enter into and participate in any WTA
Tournament. iv. Compliance with these conditions may be monitored by
testing, which testing shall be administered in WTA’s sole discretion
based on standards of reasonableness and fairness and in consultation
with physicians and medical experts. In the event of non-compliance,
the player’s eligibility for competition in WTA Tournaments will be
suspended for twelve (12) months.
B) A player who transitions from female to male (also known as a trans male (FTM)) is not eligible
to enter into or participate in any WTA Tournaments if he is undergoing hormonal treatment
for gender transition.
The policy in the ATP is less clear as it's not specifically addressed in the way the WTA addresses the matter (that I could find). This is addressed in "Entries" in rule 7.02 of the 2017 ATP Official Rulebook:
Gender / Age Limitation
ATP will use the age of a player on the first (1st) day of the qualifying competition.
Male players age sixteen (16) or older are not limited in the number of tournaments
they may enter.
However, in the Foreword under the "Exhibits" section, it states:
Note: Except where otherwise stated, every reference in these Rules of Tennis to the
masculine includes the feminine gender.
I'm not sure if this means females are included as males, or because since mixed doubles rules are covered in the document, females and males are both referred to in the same manner (ie, s/he).
Transitions before puberty are recognized as the latter gender (ie, if an individual transitions from female to male before puberty, said individual will be recognized as male).
Transitions after puberty are eligible to participate in male or female competitions respectively with respect to surgery, legal gender identity, verifiable hormonal therapy, and two years after a gonadectomy.
The ITF goes into this in Appendix A of the 2017 ITF Pro Circuit Regulations:
Any individuals undergoing sex reassignment from male to female before
puberty are regarded as girls and women (female), while those
undergoing reassignment from female to male are regarded as boys and
Individuals undergoing sex reassignment from male to
female after puberty (and the converse) are eligible for participation
in female or male competitions, respectively, under the following
a. Surgical anatomical changes have been completed,
including external genitalia changes and gonadectomy.
recognition of their assigned sex has been conferred by the
appropriate official authorities.
c. Hormonal therapy appropriate for
the assigned sex has been administered in a verifiable manner and for
a sufficient length of time to minimise genderrelated advantages in
d. Eligibility should begin no sooner than two
years after gonadectomy.