I wish I could be optimistic and encourage you here, but the answer is categorically "no." I can't tell you how many times I've seen this backfire, even when the partner does try to help the associate. My golden rule is: You should never, ever tell your firm you are looking. You should be as careful during a job search as possible. Until your conflicts are cleared at your new firm, your old firm shouldn't know anything about your search.
Let's examine a few scenarios I have encountered:
You ask your partner to help you but he does so ineffectively
Several times, I have encountered associates who have asked their partners for their assistance in helping them find another job, either as an active assistant in the search or as a reference. What can happen is that time may go on and the associate doesn't find a job after several months. Then her hours dwindle because the partner worries she may ditch him on a project and leave. She then gets a bad review and gets laid off several months later.
Your partner asks you if you are looking for a new position and you confirm that you are
It feels unethical to lie especially when directly confronted about interviewing. However, I recently worked with a finance candidate who missed work once a week for three weeks (he was interviewing in New York). When the firm asked him if the absences were for interviews, he said yes, and was let go on the spot. He felt it wrong to lie, but unfortunately he paid the price and had six months with no salary and a family to support before he found a position.
You are not careful with your conversations, emails, and voicemails
It's important to be just as careful with office staff and other associates when conducting a job search. Your job search should not be shared with your coworkers. I recently worked with an associate whose secretaries gossiped that he was interviewing (discovered via a voicemail that she had access to) and the firm happened to let him go two weeks later for a trumped up reason. Another candidate asked an associate their opinion on a firm where they were interviewing and the news got back to a partner.
Further, you should NOT use your work email to conduct a search. Always use a personal email address. Also, it's best to use your cell number, not an office line.