You should share how much you have closed, if an LP asks. However, unless you have closed at least 50% of the fund, it's sub-optimal to lead with this type of information because it's unclear if you will reach your target. This is also why it's absolutely critical to set your fund size as the minimum target rather then your maximum. The timing also matters. For example, if you recently did a 1st close and you have 20% subscribed, that's a great signal to LPs. On the other hand it doesn't look great if you set your fund size at $20m and you only closed on $5m with 6 months left in your fundraising period.

If you have a lot of fundraising momentum you can create a scarcity effect that 'the train is leaving the station, get on now'. You can use this momentum to ask current LPs for additional introductions etc.. and if they are excited about the opportunity, they will comply.