It is possible if you use xp_sqlagent_enum_jobs and then convert the Job ID value to a string value using the fn_varbintohexstr function and then find a row in sysprocesses where the program name contains that value. The login_time value in sysprocesses is then the start time of the job.
For example,
CREATE TABLE #jobs
(
Job_ID uniqueidentifier,
Last_Run_Date int,
Last_Run_Time int,
Next_Run_Date int,
Next_Run_Time int,
Next_Run_Schedule_ID int,
Requested_To_Run int,
Request_Source int,
Request_Source_ID varchar(100),
Running int,
Current_Step int,
Current_Retry_Attempt int,
State int
)
INSERT INTO #jobs EXECUTE master.dbo.xp_sqlagent_enum_jobs 1 ,''
---select * from #jobs where Running =1
SELECT login_time, program_name FROM sysprocesses WHERE lower(program_name) LIKE
(SELECT '%' + master.dbo.fn_varbintohexstr(#jobs.Job_ID) + '%' FROM #jobs
WHERE Running=1)
DROP TABLE #jobs