We've asked Tesla if it can confirm the reports. The company is expected to release its production numbers for the second quarter within days, though, so it may verify (or shoot down) the claim before long.
There's no question that Tesla's production rate has been improving. It was up to 3,500 per week in mid-June (around when its tent-like assembly line launched), and the company was comfortable enough that it started manufacturing Performance variants after months of sticking to one configuration. The very first international deliveries began in May, for that matter. The question is less about the goal itself as it is the sustainability. Did Tesla need a short-term burst to get to 5,000, or is this a sustainable figure even if there were no further upgrades?
If this is a sustainable number, it'll be important for both Tesla and its customers. The EV producer has long seen that 5,000-unit weekly goal as essential to turning a profit on Model 3 production (to date, each vehicle has represented a loss), securing its long-term health and widening the number of models it can make. If you ordered the fabled $35,000 base Model 3, you may be one step closer to getting it.
Update: Elon Musk has since tweeted that Tesla produced 7,000 cars in the space of a week, although that figure is more likely to include Tesla's other vehicles in addition to the Model 3.