An immigrant is always free to leave the United States voluntarily prior to the beginning of removal proceedings.  Following conviction or admission of an aggravated felony offense that also triggers inadmissibility (e.g., as a crime involving moral turpitude), however, a noncitizen will likely be deemed inadmissible if s/he tries thereafter to re-enter the United States.

 

BIA

RELIEF - VOLUNTARY DEPARTURE - FAILURE TO POST BOND AFTER JANUARY 20, 2009
Matter of Velasco, 25 I. & N. Dec. 143 (BIA Nov. 20, 2009) (under new regulations effective January 20, 2009, a noncitizen who failed to post a required voluntary departure bond within time required subjects noncitizen to consequences for failure to depart). 8 C.F.R. 1240.25(c)(3)-(4) (2009).

VOLUNTARY DEPARTURE - ADMISSION - DEPORTED OR TURNED BACK?
Matter of Romalez, 23 I&N Dec. 423 (BIA 2002) (voluntary departure "under threat of deportation" breaks continuous physical presence for purposes of non-LPR cancellation of removal).

NOTE: The BIA and other courts distinguish between situations where the applicant makes an agreement to voluntarily depart or withdraw an application for admission in lieu of proceedings versus those situations where the applicant was turned around at the border and did not enter into a formal agreement (even if applicant for admission was temporarily detained).

Cases finding voluntary departure or withdrawal: Matter of Avilez-Nava, 23 I&N Dec. 799 (BIA 2005); Mendez-Reyes v. Atty. Gen., 428 F.3d 187 (3d Cir. 2006); Mieles-Valdez v. Ashcroft, 349 F.3d 213 (5th Cir. 2003); Flores Palomino v. Ashcroft, 354 F.3d 942 (8th Cir. 2004); Vasquez-Lpez, 343 F.3d 961 (9th Cir. 2003).

Cases finding noncitizen was informally "turned back": MoralesMorales v. Ashcroft, 384 F.3d 418 (7th Cir. 2004); Reyes-Vasquez v. Ashcroft, 395 F.3d 903 (8th Cir. 2005); Ortiz-Cornejo v. Gonzales, 400 F.3d 610 (8th Cir. 2005); Tapia v. Gonzales, 430 F.3d 997 (9th Cir. 2005); Ibarra-Flores v. Gonzales, 439 F.3d 614 (9th Cir. 2006).

Thanks to AILF Litigation Clearinghouse Newsletter.

Ninth Circuit

RELIEF - PRE-HEARING VOLUNTARY DEPARTURE
United States v. Becerril-Lopez, 541 F.3d 881 (9th Cir. Aug. 29, 2008) (immigration judge was not required to give advice of availability of pre-hearing voluntary departure, under pre-IIRAIRA law, because the relief would have been granted prior to the respondents appearance before the IJ). Note: The court gave no opinion on the possible outcome of this issue had post-IIRAIRA rules applied, which allow the IJ to grant "pre-hearing" voluntary departure at the master calendar hearing. INA 240B(a)(1).

Tenth Circuit

RELIEF " NON-LPR CANCELLATION OF REMOVAL " CONTINUOUS PHYSICAL PRESENCE REQUIREMENT
Barrera-Quintero v. Holder, 699 F.3d 1239 (10th Cir. Nov. 15, 2012) (voluntary departure under threat of deportation breaks physical presence for purposes of non-LPR cancellation of removal).